# Arbitrarily close

John A. Rock  Department of Mathematics and Statistics, Cal Poly Pomona, 3801
W Temple Ave, Pomona, CA 91768<EMAIL_ADDRESS>www.cpp.edu/ jarock This paper
is dedicated to y and B.

###### Abstract.

Mathematicians tend to use the phrase “arbitrarily close” to mean something
along the lines of “every neighborhood of a point intersects a set”. Taking
the latter phrase as a technical definition for arbitrarily close leads to an
alternative, or at least parallel, development of classical concepts in
analysis such as closure and limits in the context of metric spaces as well as
continuity, differentiation, and integration in the setting of real valued
functions on the real line. In particular, a definition of integration in
terms of arbitrarily close is present here. The corresponding integral is
distinct from and yet equivalent to the classical integrals of Riemann and
Darboux.

###### Key words and phrases:

arbitrarily close, accretion, closed, open, supremum, infimum, limit,
continuity, differentiation, integration

###### 2010 Mathematics Subject Classification:

Primary 26A03, 26A06, 26A42, 97E40; Secondary 11B05, 54A20, 97I30, 97I40,
97I50

##  1\. INTRODUCTION AND OVERVIEW

In a first calculus course, many students are confronted with the twin notions
of limit and convergence but often with technical definitions subdued or fully
ignored. Whether in terms of sequences or functions, the definition of limit
is, frankly, difficult to understand. Once these students reach their first
undergraduate course in analysis, it is not easy to develop a technical
definition of limit following an intuitive discussion along the lines of what
it means for a sequence or function to “approach” a given value, a heuristic
phrase they may have heard in a calculus course. Thus, one of the primary
motivations of this paper is to provide a formal definition of arbitrarily
close (Definition 2.2) and, in turn, the accretion of a sequence or a function
(Definitions 6.4 and 6.4, respectively) in order to—with any luck—serve as a
more intuitive but technically sound framework for undergraduate analysis.
Ultimately, the goal is to provide a development of many of the fundamental
results in analysis based on a concrete notion of what it means for a point to
be arbitrarily close to a set.

As something of an overview, several of the ideas explored in this paper
include precise versions of the following statements:

  * •

Zero is arbitrarily close to the set of positive and the set of negative real
numbers. See Lemma 2.6.

  * •

A sequential limit is arbitrarily close to its sequence. See Theorem 2.9.

  * •

A closed set contains the points arbitrarily close to the set. See Definition
4.1 and Theorem 4.4.

  * •

A functional limit is arbitrarily close to the range of the function. See
Theorem 6.12.

  * •

A derivative is arbitrarily close to the set of difference quotients. See
Corollary 7.7.

  * •

An integral is arbitrarily close to the sets of upper and lower sums (which
are more general than Darboux sums here). See Definition 8.3 and Theorem 8.8.

The next section develops the main topic of the paper: A formal definition of
the phrase arbitrarily close.

##  2\. THE DEFINITION OF ARBITRARILY CLOSE

What is meant—mathematically—when two objects are said to be arbitrarily
close? To set the stage for a technical definition of this concept, consider
the definition of a metric space which allows for a specific meaning for the
distance between pairs of points.

######  Definition 2.1.

A metric space is a nonempty set $X$ paired with a function $d$, denoted by
$(X,d)$, where $d:X\times X\to\R$ such that:

  * (i)

$d(x,y)\geq 0$ for every pair of points $x$ and $y$ in $X$;

  * (ii)

$d(x,y)=0$ if and only if $x=y$;

  * (iii)

$d(x,y)=d(y,x)$ for every pair of points $x$ and $y$ in $X$; and

  * (iv)

for any three points $x,y,$ and $z$ in $X$ it follows that

(2.1) $\displaystyle d(x,z)+d(z,y)\leq d(x,y).$

Inequality (2.1) is known as the triangle inequality. Given a pair of points
$x$ and $y$ in $X$, the nonnegative real number $d(x,y)$ is called the
distance between $x$ and $y$.

The notion of distance provided by a metric space allows for the consideration
of the more subtle notion of how close a point is to a set. In particular,
perhaps the following definition does justice to intuition regarding what it
might mean for a point to be arbitrarily close to a set.

######  Definition 2.2.

Let $(X,d)$ be a metric space with $y\in X$ and $B\subseteq X$. The point $y$
is said to be arbitrarily close to $B$, or $B$ is arbitrarily close to $y$, if
for every $\varepsilon>0$ there is a point $x\in B$ such that
$d(x,y)<\varepsilon$. In this case, $y\operatorname{acl}B$ is
written.111Thanks to Berit Givens for the suggesting this notation. In the
case where some $z\in X$ is not arbitrarily close to $B$,
$z\operatorname{nacl}B$ is written.

In a metric space $(X,d)$, the $\varepsilon$-neighborhood centered a point $c$
of a given radius (or “error”) $\varepsilon>0$ is denoted by
$V_{\varepsilon}(c)$ and defined by

(2.2) $\displaystyle V_{\varepsilon}(c)=\\{x\in X:d(x,c)<\varepsilon\\}.$

Thus, $y\operatorname{acl}{B}$ if and only if for every $\varepsilon>0$ it
follows that $V_{\varepsilon}(c)\cap B\neq\varnothing$, where $\varnothing$
denotes the empty set.

In the context of the real line equipped with its standard metric $d_{\R}$,
$y\in\R$ is arbitrarily close $B\subseteq\R$ if and only if for every
$\varepsilon>0$ there is a point $x\in B$ such that

(2.3) $\displaystyle d_{\R}(x,y)=|x-y|<\varepsilon,\qquad\textnormal{or
equivalently,}\qquad y-\varepsilon<x<y+\varepsilon.$

The right side of Equation (2.3) just above proves to be useful in the
development of the calculus on the real line throughout the paper.

######  Remark 2.3.

A definition of arbitrarily close also holds for topological spaces: In a
topological space $X$, a point $y\in X$ is said to be arbitrarily close to a
set $B\subseteq X$ if every neighborhood of $y$ (that is, every open set
containing $y$) contains an element of $B$. By classical results in topology,
this interpretation is equivalent to Definition 2.2 in the setting of a metric
topology.

Definition 2.2 immediately yields the following fact: Points in a set are
arbitrarily close to the set.

######  Lemma 2.4.

Let $(X,d)$ be a metric space with $y\in X$ and $B\subseteq X$. If $y\in B$,
then $y$ is arbitrarily close to $B$.

###### Proof.

Assume $y\in B$ and let $\varepsilon>0$. Choosing $x=y$ yields
$d(x,y)=0<\varepsilon$. ∎

Lemma 2.4 immediately reveals the fact that in a metric space, a point which
is arbitrarily close to a set is not necessarily an accumulation point of the
set. See Definition 6.1 below. Still, these concepts are deeply related and
the foundation of their relationship is most brightly illuminated by Theorem
2.9 below.

Another basic result stemming from Definition 2.2 is the following lemma.

######  Lemma 2.5.

Let $(X,d)$ be a metric space and let $x,y\in X$. Then $x=y$ if and only if
$x$ is arbitrarily close to $\\{y\\}$ (or vice versa).

###### Proof.

Let $x=y$. For every $\varepsilon>0$, $d(x,y)=0<\varepsilon$. Thus
$x\operatorname{acl}\\{y\\}$.

Now, suppose $x\neq y$. Then $0<\varepsilon_{0}=d(x,y)/2\leq d(x,y)$. Thus
$x\operatorname{nacl}\\{y\\}$. ∎

A intuitive and fairly useful fact is that the only real number arbitrarily
close to both positive and negative real numbers is zero. To this end, let
$\R^{+}=(0,\infty)$ and $\R^{-}=(-\infty,0)$.

######  Lemma 2.6.

In the metric space $\R$ equipped with its standard metric $d_{\R}$, $\ell=0$
if and only if $\ell\operatorname{acl}\R^{+}$ and
$\ell\operatorname{acl}\R^{-}$.

###### Proof.

First, assume $\ell=0$ and let $\varepsilon>0$. Then $\varepsilon/2\in\R^{+}$,
$-\varepsilon/2\in\R^{-}$, and

(2.4) $\displaystyle-\varepsilon<$
$\displaystyle-\varepsilon/2<0<\varepsilon/2<\varepsilon.$

Therefore $\ell\operatorname{acl}\R^{+}$ and $\ell\operatorname{acl}\R^{-}$.

Next, without loss of generality, assume $\ell>0$ and let $y<0$. Then

(2.5) $\displaystyle y<0<\ell/2<\ell.$

Since $\varepsilon_{0}=\ell/2>0$, it follows that
$d_{\R}(\ell,y)=|\ell-y|>\ell/2$, and therefore $\ell$ is not arbitrarily
close to $\R^{-}$. As similar argument shows no negative number is arbitrarily
close to $\R^{+}$. ∎

In the next example and throughout the paper, the set of positive integers is
denoted by $\mathbb{N}$.

######  Example 2.7.

Consider the open interval $I=(0,3140)\subseteq\R$. Even though the real
number $\ell=3140$ is not an element of $I$, $\ell\operatorname{acl}{I}$ since
every $\varepsilon$-neighborhood of $3140$ contains an element of $(0,3140)$.
More concretely, for any $\varepsilon>0$ some $x\in I$ can be found where $x$
is less than $3140$ but greater than $3140-\varepsilon$. For instance, choose
$x=\max\\{3140-\varepsilon/2,3139\\}$. Then it follows that both that $x$ is
in $I$ and

(2.6) $\displaystyle d_{\R}(x,\ell)=|x-\ell|$
$\displaystyle=|x-3140|\leq\varepsilon/2<\varepsilon.$

Since a suitable $x$ in $I$ can be found for each error $\varepsilon>0$, a
sequence in $I$ can be found that provides a countably infinite collection of
elements as close to $\ell$ as desired. In particular, $x_{n}=3140-(1/n)$ is
in $I$ for each $n\in\mathbb{N}$ and for a given $\varepsilon>0$, $n$ can be
taken large enough to get

(2.7) $\displaystyle d_{\R}(x_{n},\ell)=|x_{n}-\ell|$
$\displaystyle=|3140-(1/n)-3140|=1/n<\varepsilon.$

The argument just above indicates how deeply the notions of arbitrarily close
and limit are related to one another. The definition of limit is arguably the
most difficult for a student of undergraduate mathematics to understand. See
[Abbott, Definition 2.2.3, p.43] but also [Barnes, Borzellino, Seager] for
some other interesting and effective approaches to teaching real analysis at
the undergraduate level which are, in part, designed to address this
difficulty. Also see [Cornu] for a thorough discussion of the challenges that
come with teaching convergence and limits. Hopefully, by first working through
the definition of and results stemming from arbitrarily close, students will
be better prepared the for challenge of understanding convergence.

To establish the deep connection between arbitrarily close and limit of a
sequence, a definition for the latter is required.

######  Definition 2.8.

Let $(X,d)$ be a metric space and for each positive integer $n$ let $x_{n}\in
X$. The sequence $(x_{n})$ converges to a point $\ell$, also written
$\lim_{n\to\infty}x_{n}=\ell$, if for every $\varepsilon>0$ there is a
positive integer $N$ such that for any $n\geq N$,

(2.8) $\displaystyle d(x_{n},\ell)$ $\displaystyle<\varepsilon.$

In this case $\ell$ is called the limit of the sequence $(x_{n})$.

In other words, $\lim_{n\to\infty}x_{n}=\ell$ means that the sequence
$(x_{n})$ is arbitrarily close to its limit $\ell$ and the sequence can be cut
off at some index $N$ so that eventually (that is, for all $n\geq N$) all of
the terms $x_{n}$ that appear in the sequence after the cutoff $N$ are as
close to $\ell$ as desired.

A considerable difference between the definition of arbitrarily close and that
of convergence (or limit) is the number of quantifiers. This distinction is
perhaps the most important reason to introduce a formal definition of
arbitrarily close before dealing with convergence. Doing so allows for a
breakdown and introduction of convergence in a couple of steps. With
arbitrarily close, one may first focus on the meaning of the quantified
statement “for every $\varepsilon>0\ldots$” along with the notion of closeness
given by “$d(x_{n},\ell)<\varepsilon$”. From there, one may tackle the subtle
multiply-quantified phrase “there is some $N\in\mathbb{N}$ such that for any
$n\geq N\ldots$” appearing in the middle of definition of convergence. The
interplay between these quantified statements seems to be a substantial
logical barrier for students to overcome. It is explored further in Section 9.

The following theorem precisely establishes the deep connection between the
definitions of convergence and arbitrarily close. Basically, points
arbitrarily close to a set are limits of sequences of points from the set.

######  Theorem 2.9.

Let $(X,d)$ be a metric space with $\ell\in X$ and $S\subseteq X$. Then
$\ell\operatorname{acl}{S}$ if and only if there is a sequence $(x_{n})$ of
points in $S$ whose limit is $\ell$.

The proof of Theorem 2.9 is straightforward and relies on nothing more than
the relevant definitions. It also makes for a nice exercise at the point where
students first have to deal with the definition of convergence.

###### Proof.

Assume $\ell$ is arbitrarily close to $S$ and note that $1/n>0$ for each
positive integer $n$. So by the definition of arbitrarily close (Definition
2.2), for each $n$ there is some $x_{n}\in S$ where

(2.9) $\displaystyle d(x_{n},\ell)<1/n.$

Now, let $\varepsilon>0$. By the intuitive Archimedean Property (see [Abbott,
Theorem 1.4.2, p.21]), there is a positive integer $N$ large enough so that
$1/N<\varepsilon$. Thus, for every $n\geq N$

(2.10) $\displaystyle d(x_{n},\ell)<1/n\leq 1/N<\varepsilon.$

Therefore, $(x_{n})\subseteq S$ and $\lim_{n\to\infty}x_{n}=\ell$.

For the converse, assume there is a sequence $(x_{n})$ of points in $S$ whose
limit is $\ell$ and let $\varepsilon>0$. By the definition of convergence
(Definition 2.8), there is some positive integer $N$ such that $x_{N}$ is in
$S$ and $d(x_{N},\ell)<\varepsilon$. Therefore, $\ell\operatorname{acl}{S}$. ∎

Be careful. Theorem 2.9 does not say that if a sequence (or rather, the range
of a sequence) is arbitrarily close to a given real number, then the limit
exists and is equal to the given number. In Example 2.10, the range of the
sequence $(b_{n})$ is arbitrarily close to two distinct real numbers
$\ell_{1}=3141\neq 3139=\ell_{2}$ and, as it turns out, neither can be the
limit due to the uniqueness of limits. Ultimately, the limit of $(b_{n})$ does
not exist.

######  Example 2.10.

Suppose $b_{n}=1/n+3140+(-1)^{n}$ for each $n\in\mathbb{N}$. The range
$S=\\{b_{n}:n\in\mathbb{N}\\}\subseteq\R$ of the sequence $(b_{n})$ is
arbitrarily close to both $\ell_{1}=3141$ and $\ell_{2}=3139$. Note that
neither $\ell_{1}=3141$ nor $\ell_{2}=3139$ is in $S$, yet for any
$\varepsilon>0$ there is a positive integer $k$ large enough to so that both

(2.11) $\displaystyle|b_{2k}-\ell_{1}|$
$\displaystyle=\left|\frac{1}{2k}+3140+(-1)^{2k}-3141\right|=\frac{1}{2k}<\varepsilon$

and

(2.12) $\displaystyle|b_{2k+1}-\ell_{2}|$
$\displaystyle=\left|\frac{1}{2k+1}+3140+(-1)^{2k+1}-3139\right|=\frac{1}{2k+1}<\varepsilon.$

Thus, the set $S$ is arbitrarily close to both $\ell_{1}=3141$ and
$\ell_{2}=3139$. Be careful. Neither $3141$ nor $3139$ is the limit of
$(b_{n})$ (and neither is in the set $S$). It can be proven that the limit of
a sequence is unique, so neither $\lim_{n\to\infty}b_{n}=3141$ nor
$\lim_{n\to\infty}b_{n}=3139$. See the classical result [Abbott, Theorem
2.2.7, p.46], where the proof is left as an exercise and makes use of Lemma
2.5.

Before moving on to more connections between the formal definition of
arbitrarily close and classical results in analysis, consider the following
example of a spiral in the complex plane which is arbitrarily close to, but
does not contain points on, the unit circle.

######  Example 2.11.

Let $f:[0,\infty)\to\C$ be given by $f(t)=te^{it}/(t+1)$ and for each $w\in\C$
let $|w|$ denote the modulus of $w$. Since

(2.13) $\displaystyle|f(t)|=\frac{t}{t+1}<1$

for every $t\in[0,\infty)$, the range of $f$ is a spiral in contained
$V_{1}(0)$, the open unit disk centered at the origin. Let $S$ denote the
spiral given by the range of $f$. It turns out that every complex number on
the unit circle centered at the origin is arbitrarily close to $S$ even though
the spiral $S$ and this circle have no points in common. Let $z=e^{i\theta}$
where $\theta\in[0,2\pi)$ and for each $n\in\mathbb{N}$ let
$z_{n}=f(\theta+2\pi n)$. Then $z$ is the point on the unit circle with
argument $\theta$ and each $z_{n}$ is a point in the intersection of the
spiral $S$ and the ray connecting the origin to $z$. Now, let $\varepsilon>0$
and let $d_{\C}$ denote the standard metric on $\C$. Then for large enough
$n$,

(2.14) $\displaystyle d_{\C}(z_{n},z)=|z_{n}-z|<\varepsilon.$

Thus $z\operatorname{acl}{S}$.

With such a deep connection between arbitrarily close and the vital notion of
convergence of sequences now established, it should come as no surprise that
many of the concepts explored in analysis can be discussed in terms of points
arbitrarily close to sets. The next section explores a way in which the
definition of arbitrarily close provided in Definition 2.2 allows for a more
intuitive derivation of the supremum and infimum of a bounded set of real
numbers.

##  3\. AN ALTERNATIVE DEFINITION OF SUPREMUM

Consider the following definition for the maximum of a set of real numbers.

######  Definition 3.1.

A real number $b$ is an upper bound of a set $S\subseteq\R$ if $x\leq b$ for
every $x\in S$. A real number $M$ is the maximum of $S$ if

  * (i)

$M$ is an upper bound for $S$, and

  * (ii)

$M\in S$.

In this case, $M=\max{S}$ is written. A similar definition holds for minimum.

The closed interval $F=[0,3140]$ has a maximum given by $\max{F}=3140$, but
what about the open interval $I=(0,3140)$? Intuition might suggest that if
this set $I$ were to have a maximum, it would certainly be 3140. The problem
is that while 3140 is an upper bound for $I$, it is not an element of $I$ and
no other real number satisfies both properties (i) and (ii) from the
definition of maximum. Thus, the maximum of $I$ does not exist.

Still, the real number $\ell=3140$ plays a special role with regard to the set
$I=(0,3140)$. The following alternative definition for the supremum shows it
be like a maximum except it is not necessarily an element of the given set.
Simply put, the supremum of a set is the upper bound arbitrarily close to the
set.

######  Definition 3.2.

A real number $u$ is the supremum of a set $S\subseteq\R$ if

  * (i)

$u$ is an upper bound for $S$, and

  * (ii)

$u\operatorname{acl}S$.

In this case, $u=\sup S$ is written.

An analogous statement holds for the definition of infimum which is denoted by
$\inf S$ and defined in terms of lower bounds of $S$. Basically, the infimum
of $S$ is the lower bound arbitrarily close to $S$.

Note that in Definition 3.1, part (ii) requires an upper bound $M$ to be in
the set in order to have $M=\max S$. In part (ii) of Definition 3.2, the
corresponding property is relaxed slightly in that an upper bound $u$ need
only be arbitrarily close to the set in order to have $u=\sup S$.

As seen in Example 2.7, the real number 3140 is arbitrarily close to open
interval $I=(0,3140)$. By the definition of an open interval it follows that

(3.1) $\displaystyle I=(0,3140)=\\{x\in\R:0<x<3140\\},$

so 3140 is also an upper bound for $I$. Therefore, $\sup I=3140$.

To see that the definition of supremum provided by Definition 3.2 is
equivalent to a more classical definition employed by Abbott (see [Abbott,
Definition 1.3.2, p.15])—that is, the supremum of a set is its least upper
bound—look to a result provided by Abbott (see [Abbott, Lemma 1.3.8, p.17]).
The corresponding lemma is stated here for convenience. Essentially, it says
an upper bound is the supremum of a set when it is arbitrarily close to the
set.

######  Lemma 3.3.

Assume $u\in\R$ is an upper bound for a set $A\subseteq\R$. Then $u=\sup A$ if
and only if for every choice of $\varepsilon>0$, there exists some element
$a\in A$ satisfying $u-\varepsilon<a$.

Assuming $u$ is an upper bound for $A$, $\varepsilon>0$, and there exists some
element $a\in A$ satisfying $u-\varepsilon<a$, it follows that

(3.2) $\displaystyle u-\varepsilon<a\leq u<u+\varepsilon,$

and therefore $d_{\R}(a,u)=|a-u|<\varepsilon$. Hence, $u=\sup A$ and $u$ is an
upper bound arbitrarily close to $A$. As a result, Definition 3.2 serves as a
viable alternative for the definition of supremum.

Analogous results hold for infimum and the proofs follow readily, as with the
following corollary.

######  Corollary 3.4.

Suppose $S\subseteq\R$ with $u=\sup{S}$. Then there is a sequence $(x_{n})$
contained in $S$ whose limit is $u$.

###### Proof.

The statement follows readily from Theorem 2.9 and Lemma 3.3. ∎

The technical definition of arbitrarily close in Definition 2.2 not only
provides an alternative approach to defining integration and understanding
supremum and infimum, it also immediately connects to the fundamental aspects
of metric spaces, especially closed sets.

##  4\. A CLOSED-MINDED APPROACH TO ANALYSIS

The setting and development of closed sets (and open sets) described here is
restricted to metric spaces, but many of the results hold in the more general
context of topological spaces. More importantly, establishing a solid notion
of closed sets through the definition of arbitrarily close in Definition 2.2
allows for an alternative foundation for many results in calculus.

Throughout this section, let $X$ denote a metric space with metric $d$.

######  Definition 4.1.

Let $B\subseteq X$. The closure of $B$, denoted by $\overline{B}$, is the set
of points arbitrarily close to $B$. Thus,

(4.1) $\displaystyle\overline{B}=\\{x\in X:x\operatorname{acl}B\\}=\\{x\in
X:\,\forall\,\varepsilon>0,V_{\varepsilon}(x)\cap B\neq\varnothing\\}.$

A set $F\subseteq X$ is closed if $F$ contains the points in $X$ arbitrarily
close to $F$.

Definition 4.1 immediately yields a fundamental fact about closed sets.

######  Lemma 4.2.

The intersection of a collection of closed sets is closed.

###### Proof.

Let $\\{F_{\alpha}\\}$ denote a collection of closed sets in some metric space
and suppose $y$ is arbitrarily close to $\cap_{\alpha}F_{\alpha}$. Let
$\varepsilon>0$ and consider the $\varepsilon$-neighborhood of
$V_{\varepsilon}(y)$. Then there is some $x\in(\cap_{\alpha}F_{\alpha})\cap
V_{\varepsilon}(y)$ which means $x\in F_{\alpha}\cap V_{\varepsilon}(y)$ for
each $\alpha$. Since $\varepsilon$ was chosen arbitrarily and each
$F_{\alpha}$ is closed, it follows that $y\in F_{\alpha}$ for each $\alpha$,
thus $y\in\cap_{\alpha}F_{\alpha}$. Therefore, $\cap_{\alpha}F_{\alpha}$ is
closed. ∎

Any point $z$ which is not in a closed set $F$ is by Definition 4.1 is not
arbitrarily close to $F$. Hence, there is an $\varepsilon$-neighborhood
$V_{\varepsilon}(z)$ of $z$ which does not intersect $F$. This is precisely a
characterizing property—therefore a defining property—of points in an open set
(cf. [Abbott, Definition 3.2.1, p.88]). To this end, the complement of a set
$S$ in a metric space $X$ is denoted by $S^{c}$ and given by

(4.2) $\displaystyle S^{c}=X\setminus S=\\{x\in X:x\notin S\\}.$

Also, a point $y$ is not arbitrarily close to a set $S$ if there is some
$\varepsilon_{0}>0$ such that every $x\in S$ satisfies

(4.3) $\displaystyle d(x,y)\geq\varepsilon_{0}.$

In other words, for some positive distance $\varepsilon_{0}$, every $x\in S$
is at least $\varepsilon_{0}$ away from $y$. Equivalently,
$y\operatorname{nacl}S$ if and only if there is some
$\varepsilon_{0}$-neighborhood of $y$ that does not intersect $S$. That is,
$V_{\varepsilon_{0}}(y)\cap S=\varnothing$ for some $\varepsilon_{0}>0$.

######  Definition 4.3.

A set $O\subseteq X$ is open if for every $a\in O$ there is an
$\varepsilon$-neighborhood of $a$ contained in $O$ (that is,
$V_{\varepsilon}(a)\subseteq O$ for some $\varepsilon>0$).

The key topological connection between open and closed sets of a metric space
readily follows (cf. [Abbott, Theorem 3.2.13, p.92]).

######  Theorem 4.4.

A set $O\subseteq X$ is open if and only if $O^{c}$ is closed.

A proof very similar to the one presented here was created by a student as she
prepared for a final exam in the summer of 2019. In particular, she used the
language of arbitrarily close and preferred this approach over the one used in
[Abbott, Theorem 3.2.13, p.92].

###### Proof.

Assume $O$ is open and suppose $y$ is arbitrarily close to $O^{c}$. By way of
contradiction, assume $y\in O$. Since $O$ is open, there is an
$\varepsilon_{0}$-neighborhood of $y$ contained in $O$. This means every $x\in
O^{c}$ lives outside of this $\varepsilon_{0}$-neighborhood of $y$, thus $x$
is at least a positive distance $\varepsilon_{0}$ away from $y$. Hence, $y$ is
not arbitrarily close to $O^{c}$, a contradiction. Therefore, $O^{c}$ is
closed.

For the converse, assume $O^{c}$ is closed and let $z\in O$. Since $O^{c}$
contains all points arbitrarily close to $O^{c}$, $z$ is not one of them.
Since $z$ is not arbitrarily close to $O^{c}$, there must be some
$\varepsilon_{z}>0$ such that $V_{\varepsilon_{z}}(z)\subseteq O$. Therefore,
$O$ is open. ∎

The following is just a concrete example illustrating Theorem 4.4.

######  Example 4.5.

Consider the closed interval $F=[0,3140]$ and let $w=3141$. Also, consider the
positive distance $\varepsilon_{0}=1/10$. Then for every $x\in F$ it follows
that

(4.4) $\displaystyle|x-3141|\geq 1/10=\varepsilon_{0}.$

As a result, the neighborhood $V_{1/10}(3141)=(3141-1/10,3141+1/10)$ contains
$w$ and is completely away from $F$ in the sense that

(4.5) $\displaystyle[0,3140]\cap(3141-1/10,3141+1/10)=\varnothing.$

Now consider the complement $F^{c}=\R\setminus
F=(-\infty,0)\cup(3140,\infty)$. Every element in $\R\setminus F$ comes with
an $\varepsilon$-neighborhood that is also contained in $\R\setminus F$.
Specifically, for each $z\in\R\setminus F$, define $\varepsilon_{z}$ to be the
shorter of the distances between $z$ and the endpoints of $F=[0,3140]$. So,
$\varepsilon_{z}=\min\\{|z-0|,|z-3140|\\}$ and $\varepsilon_{z}>0$ since
$z\neq 0$ and $z\neq 3140$. Then

(4.6) $\displaystyle V_{\varepsilon_{z}}(z)\cap[0,3140]=\varnothing.$

Therefore, $V_{\varepsilon_{z}}(z)\subseteq\R\setminus F$.

That is, every element in the complement of the closed interval $F$ comes with
an $\varepsilon$-neighborhood that is also contained in the complement
$F^{c}=\R\setminus F$, which is open by Theorem 4.4.

Metric spaces provide a natural setting for a notion of boundedness.

######  Definition 4.6.

Let $(X,d)$ denote a metric space. A set $A\subseteq X$ is said to be bounded
if there is a point $z$ in $X$ a nonnegative real number $u$ such that

(4.7) $\displaystyle d(z,x)\leq u\qquad\textnormal{for all }x\in A.$

Thanks to the triangle inequality (2.1), the choice of the point $z$ as above
has no effect on whether or not a given set is bounded. In the context of the
real line $\R$ or the complex plane $C$ equipped with their standard metrics,
the choice $z=0$ is made throughout the paper.

######  Lemma 4.7.

Suppose $A$ is bounded in a metric space $(X,d)$ with $z$ in $X$ and $u\geq 0$
such that $d(z,x)\leq u$ for every point $x$ in $A$. Then $\overline{A}$ is
bounded and $d(z,y)\leq u$ for every point $y$ in $\overline{A}$.

###### Proof.

Suppose $A$ is bounded with $z$ in $X$ and $u\geq 0$ such that $d(z,x)\leq u$
for every point $x$ in $A$. Let $y\in\overline{A}$ and $\varepsilon>0$. Since
$y\operatorname{acl}A$, there is some $a$ in $A$ such that
$d(a,y)<\varepsilon$. Paired with the triangle inequality (2.1), it follows
that

(4.8) $\displaystyle d(z,y)\leq d(z,a)+d(a,y)<u+\varepsilon.$

Since $\varepsilon>0$ is arbitrary, $\overline{A}$ is bounded and $d(z,y)\leq
u$ for every point $y$ in $\overline{A}$. ∎

The boundary of a set is another topological concept that readily lends itself
to an alternative—and perhaps more intuitive—definition. For comparison, see
[AdamsFranzosa, Definition 2.13, p.65] but also the discussions on page 9
where the phrase “arbitrarily close” is used but not defined, and page 65
regarding “points that lie close to both the inside and the outside of the
set”.

######  Definition 4.8.

The boundary of a set $S\subseteq X$, denoted by $\partial S$, is the set of
points arbitrarily close to both $S$ and its complement $S^{c}$.

Again, this definition of boundary is intuitive. Points on the boundary of a
set are near both the set and the complement of the set. Such points live in
the closure, but not the interior, of the set.

######  Definition 4.9.

Let $S\subseteq X$. The interior of $S$, denoted by $\mathring{S}$, is the
union of all the open sets contained in $S$. That is, $\mathring{S}$ is the
largest open set contained in $S$.

The next theorem follows from the fact that each point in an open set comes
with an $\varepsilon$-neighborhood which is also contained in the set.

######  Theorem 4.10.

For any set $S\subseteq X$, $\partial{S}=\overline{S}\setminus\mathring{S}$.

###### Proof.

Assume $y\in\partial{S}$. Then $y\operatorname{acl}{S}$ and
$y\operatorname{acl}{S^{c}}$. So, $y\in\overline{S}$ and for every
$\varepsilon>0$ it follows that $V_{\varepsilon}(y)\cap S^{c}\neq\varnothing$.
Hence, no $\varepsilon$-neighborhood of $y$ is contained in $S$, so
$y\notin\mathring{S}$.

For the converse, assume $y\in\overline{S}\setminus\mathring{S}$. Since
$y\in\overline{S}$ it follows that $y\operatorname{acl}{S}$. But since
$y\notin\mathring{S}$, for every $\varepsilon>0$ it follows that
$V_{\varepsilon}(y)\cap S^{c}\neq\varnothing$. Thus
$y\operatorname{acl}{S^{c}}$ as well. ∎

To conclude this section, consider the example of the topologist’s sine curve
realized in the complex plane.

######  Example 4.11.

Let $S\subseteq\C$ be the range of the function $g:\R^{+}\to\C$ given by

(4.9) $\displaystyle g(t)=t+i\sin\left(\frac{1}{t}\right)$

and let $T=\\{iy:-1\leq y\leq 1\\}$, which is the closed line segment
connecting $-i$ and $i$ in the complex plane. Even though $S\cap
T=\varnothing$, it follows that $T\subseteq\partial{S}$. Indeed, given any
point in $T$ there is a convergent sequence of points in the range of $g$ and
another convergent sequence outside of the range of $g$ whose limits are the
given point.

Let $z_{0}\in T$ and $\varepsilon>0$. Then $z_{0}=iy_{0}$ for some
$y_{0}\in[-1,1]$. There is some $\omega_{0}\in[0,2\pi)$ such that
$\sin\omega_{0}=y_{0}$. For each $n\in\mathbb{N}$, define
$t_{n}=1/(\omega_{0}+2\pi n)$. Thus, for any $\varepsilon>0$ and $n\geq N$
where

(4.10) $\displaystyle
N>\frac{1}{2\pi}\left(\frac{1}{\varepsilon}-\omega_{0}\right),$

it follows that

(4.11) $\displaystyle
d_{\C}(g(t_{n}),z_{0})=\left|g(t_{n})-z_{0}\right|=t_{n}=\frac{1}{\omega_{0}+2\pi
n}\leq\frac{1}{\omega_{0}+2\pi N}<\varepsilon.$

Therefore, $\lim_{n\to\infty}g(t_{n})=z_{0}$. By Theorem 2.9, it follows that
$z_{0}\operatorname{acl}{S}$.

Now consider the sequence $(z_{0}-1/(\omega_{0}+2\pi n))$. Each point in this
sequence is in $S^{c}$. For $\varepsilon$ and $n\geq N$ as above it follows
that $z_{0}-1/(\omega_{0}+2\pi n)$ is in $V_{\varepsilon}(z_{0})\cap S^{c}$.
Therefore, $z_{0}\operatorname{acl}{S^{c}}$ as well. By Theorem 4.10 it
follows that $z_{0}\in\partial{S}$.

Sequences have already played a prominent role in the development of the paper
thus far, but the following section explores properties of sequences through
the lens of arbitrarily close by taking closures of tails.

##  5\. ACCRETION OF SEQUENCES

What is meant when a sequence is said to “approach” a given point? In calculus
texts, a phrase such as “the sequence $(x_{n})$ approaches $\ell$” may be
taken to mean $\lim_{n\to\infty}x_{n}=\ell$. But such a correspondence merely
ties a vague phrase to a very technical statement, leaving a lot to be
desired. The setting provided by gathering all points arbitrarily close to a
given sequence provides an intermediate step towards understanding limits of
sequences.

######  Definition 5.1.

Let $(x_{n})$ be a sequence of points in a metric space. The accretion of
$(x_{n})$, denoted by $A((x_{n}))$, is defined by

(5.1) $\displaystyle
A((x_{n}))=\bigcap_{N\in\mathbb{N}\cup\\{0\\}}\overline{\\{x_{N+n}:n\in\mathbb{N}\\}}.$

That is, the accretion of a sequence is the intersection of the closure of the
tails of the sequence. To foster intuition, it helps to unpack Definition 5.1
as follows:

  1. (1)

First, for each $N\in\mathbb{N}\cup\\{0\\}$, the $N$-tail (or simply tail) of
the sequence $(x_{n})$ is the subsequence $(x_{N+n})$ whose first term is
$x_{N+1}$ and which is a copy of the sequence $(x_{n})$ thereafter.

  2. (2)

Taking the closure of each $N$-tail, denoted by
$\overline{\\{x_{N+n}:n\in\mathbb{N}\\}}$ above, picks up all points
arbitrarily close to the $N$-tail (technically, all points arbitrarily close
to the range of the corresponding subsequence).

  3. (3)

Finally, the intersection of the closures of all $N$-tails, denoted above by

(5.2)
$\displaystyle\bigcap_{N\in\mathbb{N}\cup\\{0\\}}\overline{\\{x_{N+n}:n\in\mathbb{N}\\}},$

amounts to gathering all points arbitrarily close to the sequence $(x_{n})$
and truncating terms which are not arbitrarily close to each of the $N$-tails.

There is an immediate corollary of Lemma 4.2 since $A((x_{n}))$ is an
intersection of closed sets.

######  Corollary 5.2.

The accretion of a sequence is closed.

The structure of the accretion of a sequence can vary considerably, as the
following examples show.

######  Example 5.3.

Consider the sequence of real numbers given by $(n)$, the enumeration of the
natural numbers. Then for each $N\in\mathbb{N}\cup\\{0\\}$ it follows that

(5.3) $\displaystyle
A((n))=\bigcap_{N\in\mathbb{N}\cup\\{0\\}}\overline{\\{N+n:n\in\mathbb{N}\\}}=\bigcap_{N\in\mathbb{N}\cup\\{0\\}}\\{N+1,N+2,\ldots\\}=\varnothing.$

Thus, it is possible for the accretion of a sequence to be empty.

######  Example 5.4.

Let $(z_{n})$ be the sequence of complex numbers with terms defined for each
$n\in\mathbb{N}$ by $z_{n}=i^{n}(n-1)/n=i^{n}-(i^{n}/n)$. Then

(5.4) $\displaystyle A((z_{n}))=$
$\displaystyle\bigcap_{N\in\mathbb{N}\cup\\{0\\}}\overline{\\{i^{N+n}-(i^{N+n}/(N+n)):n\in\mathbb{N}\\}}$
(5.5) $\displaystyle=$ $\displaystyle\,\\{i^{k}:k=0,1,2,3\\}=\\{1,i,-1,-i\\}.$

Indeed, for each $i^{k}$ where $k=0,1,2,3$ and each $\varepsilon>0$, there is
a natural number $m$ large enough so that

(5.6) $\displaystyle|z_{4m+k}-i^{k}|=1/(4m+k)<\varepsilon.$

Thus, for each $k=0,1,2,3$ and each $N\in\mathbb{N}\cup\\{0\\}$ it follows
that

(5.7) $\displaystyle
i^{k}\operatorname{acl}\\{i^{N+n}-(i^{N+n}/(N+n)):n\in\mathbb{N}\\}.$

No other complex number is arbitrarily close to each $N$-tail of the sequence
$(z_{n})$.

######  Example 5.5.

Let $(q_{n})$ be an enumeration of the rational numbers in the closed unit
interval $[0,1]$. That is, there is a bijection
$f:\mathbb{N}\to\mathbb{Q}\cap[0,1]$ where $f(n)=q_{n}$ for each natural
number $n$. Then by the density of the rational numbers in the real numbers,
for each $N\in\mathbb{N}\cup\\{0\\}$ it follows that
$\overline{\\{q_{N+n}:n\in\mathbb{N}\\}}=[0,1]$ despite the fact that the
first $N$ terms have been left out of corresponding tail of the sequence.
Therefore, $A((q_{n}))=[0,1]$. Note that this result is stronger than the fact
that $\overline{\mathbb{Q}\cap[0,1]}=[0,1]$.

The next example shows that care must be taken when connecting the notions of
accretion and limit of a sequence. It is not enough to suggest that the
accretion need only be singleton in order for the limit of the sequence to
exist.

######  Example 5.6.

Let $(b_{n})$ be the sequence of real numbers with terms defined by $b_{n}=0$
when $n$ is odd and $b_{n}=n$ when $n$ is even. Then for each
$N\in\mathbb{N}\cup\\{0\\}$ and the corresponding $N$-tail it follows that
$\overline{\\{b_{N+n}:n\in\mathbb{N}\\}}\subseteq\\{0\\}\cup\\{N+1,N+2,\ldots\\}$.
Hence,

(5.8) $\displaystyle\\{0\\}\subseteq
A((b_{n}))\subseteq\bigcap_{N\in\mathbb{N}\cup\\{0\\}}(\\{0\\}\cup\\{N+1,N+2,\ldots\\})=\\{0\\}.$

Therefore, $A((b_{n}))=\\{0\\}$, but as an unbounded sequence $(b_{n})$ does
not converge.

The following proposition provides a feature of convergent sequences in the
context of accretion.

######  Proposition 5.7.

If a sequence of points $(x_{n})$ in a metric space $(X,d)$ converges, then
its accretion $A((x_{n}))$ is a singleton and

(5.9) $\displaystyle A((x_{n}))=\left\\{\lim_{n\to\infty}x_{n}\right\\}.$

###### Proof.

Assume $\lim_{n\to\infty}x_{n}=x$ and let $\varepsilon>0$. There is some
positive integer $K$ such that for every positive integer $k\geq K$ it follows
that $d(x_{k},x)<\varepsilon.$ So for every $N\in\mathbb{N}\cup\\{0\\}$ it
follows that

(5.10) $\displaystyle x\in\overline{\\{x_{N+n}:n\in\mathbb{N}\\}}.$

Hence, $\lim_{n\to\infty}x_{n}=x\in A((x_{n}))$.

Now suppose $y\neq x$. Then $d(x,y)>0$ and there is some $N_{1}\in\mathbb{N}$
such that for all $k\geq N_{1}$ it follows that

(5.11) $\displaystyle d(x,y)\leq
d(x,x_{k})+d(x_{k},y)<\frac{1}{2}d(x,y)+d(x_{k},y),$

and therefore,

(5.12) $\displaystyle\frac{1}{2}d(x,y)\leq d(x_{k},y).$

Thus, $y\notin\overline{\\{x_{N_{1}+n}:n\in\mathbb{N}\\}}$ and so $y\notin
A((x_{n}))$. ∎

A definition for subsequences is needed before concluding the section.

######  Definition 5.8.

Let $(x_{n})$ be a sequence of points in a metric space and let
$n_{1}<n_{2}<n_{3}<\cdots$ be a strictly increasing sequence of natural
numbers. Then the sequence

(5.13) $\displaystyle(x_{n_{1}},x_{n_{2}},x_{n_{3}},\ldots)$

is called a subsequence of $(x_{n})$ and is denoted by $(x_{n_{k}})$ where
$k\in\mathbb{N}$ indexes the subsequence. If $\ell=\lim_{k\to\infty}x_{n_{k}}$
for some subsequence $(x_{n_{k}})$ of the sequence $(x_{n})$, then $\ell$ is
called a subsequential limit of $(x_{n})$. Let $L((x_{n}))$ denote the set of
subsequential limits of $(x_{n})$.

The accretion of a sequence is the set of subsequential limits of the
sequence.

######  Proposition 5.9.

Let $(x_{n})$ be a sequence of points in a metric space. Then
$A((x_{n}))=L((x_{n}))$.

###### Proof.

This result follows from Theorem 2.9 and the fact that a subsequence is itself
a sequence. ∎

The accretion of a sequence connects directly to the classical notions of
limit superior and limit inferior for bounded sequences of real numbers.

######  Definition 5.10.

Let $(x_{n})$ be a bounded sequence of real numbers. Then the limit superior
and limit inferior of $(x_{n})$, denoted by $\limsup{x_{n}}$ and
$\liminf{x_{n}}$, respectively, are defined by

(5.14) $\displaystyle\limsup{x_{n}}=\sup
L((x_{n}))\quad\textnormal{and}\quad\liminf{x_{n}}=\inf L((x_{n})).$

The next corollary follows immediately from a string of results obtained thus
far. Essentially, the limit superior and limit inferior of a bounded sequence
of real numbers are the maximum and minimum, respectively, of the accretion of
the sequence.

######  Corollary 5.11.

If $(x_{n})$ is a bounded sequence of real numbers, then

(5.15) $\displaystyle\limsup{x_{n}}\in
L((x_{n}))\quad\textnormal{and}\quad\liminf{x_{n}}\in L((x_{n})).$

Equivalently,

(5.16)
$\displaystyle\limsup{x_{n}}=\max{A((x_{n}))}\quad\textnormal{and}\quad\liminf{x_{n}}=\min{A((x_{n}))}.$

###### Proof.

The proof follows from Proposition 5.9, Corollary 5.2, Lemma 4.7 (to ensure
the corresponding supremum and infimum exist), and Corollary 3.4 (and its
analog for infima). ∎

The section concludes with the following characterization of convergent
sequences of real numbers.

######  Corollary 5.12.

Suppose $(x_{n})$ is a sequence of real numbers. Then $\lim_{n\to\infty}x_{n}$
exists if and only if the accretion $A((x_{n}))$ is a singleton and $(x_{n})$
is bounded.

###### Proof.

If $\lim_{n\to\infty}x_{n}$ exists, then $(x_{n})$ is bounded (see [Abbott,
Theorem 2.3.2, p.49]) and by Proposition 5.7 is a singleton.

On the other hand, if $A((x_{n}))$ is a singleton and $(x_{n})$ is bounded,
then by Corollary 5.11 it follows that $\limsup{x_{n}}=\liminf{x_{n}}$ and so
$\lim_{n\to\infty}x_{n}$ exists. ∎

The notion of accretion for functions defined in the next section allows for a
parallel approach to functional limits and pointwise continuity.

##  6\. ACCRETION AND LIMITS OF FUNCTIONS

The definition of the limit of a function is another difficult concept for new
mathematicians to fully understand, as indicated in [Barnes, Borzellino,
Cornu, Seager] and as can be corroborated by just about any undergraduate
mathematics major. As such, this section is designed to face the challenge of
understanding limits and pointwise continuity of functions by providing an
alternative development in the context of accretion.

In order to align with a more classical approach to analysis, consider the
following definition.

######  Definition 6.1.

Let $(X,d)$ denote a metric space. A point $c\in X$ is an accumulation point
of a set $A\subseteq X$ if $c\operatorname{acl}{(A\setminus\\{c\\})}$. That
is, $c$ is an accumulation point of $A$ if for every $\varepsilon>0$ it
follows that $V_{\varepsilon}(\ell)\cap(A\setminus\\{c\\})\neq\varnothing$.

In other words, $c$ is an accumulation point of $A$ if every
$\varepsilon$-neighborhood of $c$ intersects $A$ at some point other than $c$.

In [Abbott, AdamsFranzosa] and other undergraduate textbooks on analysis,
accumulation points are also called “cluster points” or “limit points”.222By
definition, not all limits are limit points. Consider a constant sequence
compared to the singleton comprising the constant. For instance, see [Abbott,
Definition 3.2.4, p.89] and [AdamsFranzosa, Definition 2.7, p.60]. Such
terminology is justified since accumulation points of a set are limits of
sequences in the set. See [Abbott, Theorem 3.2.5, p.89], the statement of
which is provided in the following corollary of Theorem 2.9.

######  Corollary 6.2.

Let $(X,d)$ denote a metric space. A point $x\in X$ is an accumulation point
of a set $A\subseteq X$ if and only if $x=\lim_{n\to\infty}a_{n}$ for some
sequence $(a_{n})$ of points in $A$ with $x\neq a_{n}$ for every
$n\in\mathbb{N}$.

###### Proof.

The statement is a special case of Theorem 2.9. Points arbitrarily close to a
set are limits of sequences of points contained in the set. ∎

Here is a classical definition of the limit of a function from one metric
space to another.

######  Definition 6.3.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. Let $A$ be a subset of $X$,
let $f:A\to Y$, and let $c$ be an accumulation point of $A$. A point $\ell\in
Y$ is the limit of $f$ at $c$, written $\lim_{x\to c}f(x)=\ell$, if for every
$\varepsilon>0$ there is a response $\delta>0$ such that for every $x\in A$
where $0<d_{X}(x,c)<\delta$ it follows that $d_{Y}(f(x),f(c))<\varepsilon.$

In other words, $\lim_{x\to c}f(x)=\ell$ if for every $\varepsilon>0$ there is
some $\delta>0$ such that $x\in(V_{\delta}(c)\cap A)\setminus\\{c\\}$ implies
$f(x)\in V_{\varepsilon}(\ell)$.

Definition 6.3 is a far cry from the intuition students develop in a calculus
class when it comes to discussing the definition of a limit of a real valued
function at some real number $c$. Calculus textbooks often use a statement
such as “the limit of a function at $c$ is the value the outputs approach as
the inputs approach $c$” and related figures to help the reader understand the
situation. However, such statements leave a lot to be desired. They are
neither technically robust nor are the associated figures clearly connected to
a technical definition such as Definition 6.3. There is room for improvement.

An alternative stems from concept of the accretion of a function associated
with a point. The idea is to map neighborhoods of the point in the domain
first, before concerning ourselves $\varepsilon$-neighborhoods of the output
of the point. After all, in calculus classes our students may be taught to
develop an intuition about a function by plugging values in and seeing what
happens with the outputs.

######  Definition 6.4.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces, let $B,D\subseteq X$, and
let $f:D\to Y$. Given a set point $c\in X$, the accretion of $f$ at $c$ with
respect to $B$, denoted by $A(f,c,B)$, is the set given by

(6.1) $\displaystyle A(f,c,B)=\bigcap_{\delta>0}\overline{f(V_{\delta}(c)\cap
B)}.$

If $D\cap B=\varnothing$ or if $V_{\delta_{0}}(c)\cap B=\varnothing$ for some
$\delta_{0}>0$, then $A(f,c,B)=\varnothing$.

When first encountered, Definition 6.4 may seem to be just as opaque as the
classical definition of limit of function provided by Definition 6.3. However,
Definition 6.4 can be unpacked and connected to the intuition students (might)
develop in calculus as follows:

  1. (1)

First, for each $\delta>0$, the set of points of interest are gathered into
the set $V_{\delta}(c)\cap B$ (the points in $B$ within $\delta$ of $c$).

  2. (2)

Following intuition developed in calculus (maybe), the function is then
evaluated at these points.

  3. (3)

For the next step, which is key to the development from the material from this
point on, the closure of the image $f(V_{\delta}(c)\cap B)$—that is, the set
of points in the codomain arbitrarily close to this image—is taken for each
$\delta>0$.

  4. (4)

After taking the closures of the images for each $\delta$, the final step is
to intersect these closures over all positive $\delta$, solidifying the
intuitive notion of having “$\delta$ approaches $0$”. Taking this intersection
amounts to the intuition of keeping only points arbitrarily close to the
outputs that always remain as the inputs approach $c$.

  5. (5)

The resulting set $A(f,c,B)$, the accretion of $f$ at $c$ with respect to $B$,
is thus the set of points the outputs of $f$ approach as the inputs approach
$c$.

An immediate consequence of the definition of accretion for functions
(Definition 6.4) and follows from the fact that in a metric space,
intersections of closed sets are closed (Lemma 4.2).

######  Corollary 6.5.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces, let $B,D\subseteq X$, and
let $f:D\to Y$. Given any point $c\in X$ and any set $B\subseteq X$, the set
$A(f,c,B)$ is closed in $Y$.

For the sake of exposition, the examples and results obtained in the remainder
of this section are limited to functions to and from the real line.

The following examples illustrate the concept of accretion for functions and
show that care must be taken when comparing the definition of accretion in
Definition 6.4 with limits and continuity.

######  Example 6.6.

Consider Dirichlet’s function, the indicator function of the rationals
$\mathds{1}_{\mathbb{Q}}:\R\to\R$ given by

(6.2)
$\displaystyle\mathds{1}_{\mathbb{Q}}(x)=\begin{cases}1,&\quad\textnormal{if}\quad
x\in\mathbb{Q},\\\ 0,&\quad\textnormal{if}\quad
x\in\R\setminus\mathbb{Q}.\end{cases}$

Dirichlet’s function is a classic example of a nowhere-continuous function as
discussed in Remark 7.4 below. Due the density of the rational numbers
$\mathbb{Q}$ and irrational numbers $\R\setminus\mathbb{Q}$ in the real line
$\R$ (that is, the fact that every $\delta$-neighborhood of any real number
contains both rational and irrational numbers, see [Abbott, Theorems 1.4.3 and
1.4.4, p.22]), for each $c\in\R$ it follows that

(6.3) $\displaystyle A(\mathds{1}_{\mathbb{Q}},c,\R)=\\{0,1\\}.$

That is, the accretion of $\mathds{1}_{\mathbb{Q}}$ at any real number $c$
with respect to the domain $\R$ is the set containing $0$ and $1$.

######  Example 6.7.

Consider Thomae’s function $t:\R\to\R$ given by

(6.4) $\displaystyle t(x)=\begin{cases}1,&\quad\textnormal{if }x=0,\\\
1/n,&\quad\textnormal{if }x\in\mathbb{Q},x=m/n\textnormal{ is in reduced form
with }n>0,\\\ 0,&\quad\textnormal{if }x\in\R\setminus\mathbb{Q}.\end{cases}$

As discussed in [Abbott, p.114], the function $t$ is continuous on
$\R\setminus\mathbb{Q}$ and discontinuous at each rational number. Also,

(6.5) $\displaystyle A(t,c,\R)=\begin{cases}\\{0,1\\},&\quad\textnormal{if
}c=0,\\\ \\{0,1/n\\},&\quad\textnormal{if }c\in\mathbb{Q},c=m/n\textnormal{ is
in reduced form with }n>0,\\\ \\{0\\},&\quad\textnormal{if
}c\in\R\setminus\mathbb{Q}.\end{cases}$

######  Example 6.8.

Consider the function $f:\R\to\R$ given by

(6.6) $\displaystyle f(x)=\begin{cases}0,&\quad\textnormal{if}\quad x=0,\\\
1/x,&\quad\textnormal{if}\quad x\neq 0.\end{cases}$

This function is continuous everywhere in $\R$ except at $c=0$ where it is
locally unbounded and the limit does not exist. That is, at $c=0$ and for each
$\delta>0$ it follows that $0\in f(V_{\delta}(0)\cap\R)$ and, moreover,

(6.7)
$\displaystyle\overline{f(V_{\delta}(0)\cap\R)}=(-\infty,-1/\delta]\cup\\{0\\}\cup[1/\delta,\infty).$

The intersection of these closures have only 0 in common, so

(6.8) $\displaystyle
A(f,0,\R)=\bigcap_{\delta>0}\overline{f(V_{\delta}(0)\cap\R)}=\\{0\\}.$

Thus, even though the accretion of $f$ at $c=0$ with respect to $\R$ is a
singleton, $\lim_{x\to 0}f(x)$ does not exist and $f$ is not continuous at
$c$.

To avoid issues with existence of limits as in the previous example, the local
behavior of the function needs to be constrained somewhat.

######  Definition 6.9.

Let $D\subseteq\R$, let $f:D\to\R$, and let $c$ be an accumulation point of
$D$. The function $f$ is said to be locally bounded at $c$ if $f(D\cap
V_{\delta}(c))$ is bounded for every $\delta>0$.

######  Definition 6.10.

Let $D\subseteq\R$, let $f:D\to\R$, and let $c$ be an accumulation point of
$D$. The accretion limit of $f$ at $c$, denoted by $\operatorname*{alim}_{x\to
c}f(x)$, is said to exist if

  * (i)

$f$ is locally bounded at $c$, and

  * (ii)

$A(f,c,D\setminus\\{c\\})$ is a singleton.

In this case, $A(f,c,D\setminus\\{c\\})=\\{\operatorname*{alim}_{x\to
c}f(x)\\}$. If $c$ is not an accumulation point of $D$, then $f$ is said to
not have an accretion limit at $c$.

Note that the value of $f$ at $c$ is irrelevant in the context of accretion
limits at $c$ and, formally, it follows

(6.9) $\displaystyle A(f,c,D\setminus\\{c\\})$
$\displaystyle=\bigcap_{\delta>0}\overline{f(D\cap
V_{\delta}(c)\setminus\\{c\\})}$ (6.10)
$\displaystyle=\bigcap_{\delta>0}\overline{\left\\{f(x):x\in D\textnormal{ and
}0<|x-c|<\delta\right\\}}.$

Aligning with intuition, when the accretion limit of $f$ at $c$ exists and as
$\delta>0$ is taken to be smaller and smaller, the images $f(D\cap
V_{\delta}(c)\setminus\\{c\\})$ shrink down to the singleton containing just
$\operatorname*{alim}_{x\to c}f(x)$. Ultimately, limits and accretion limits
coincide. The following lemma provides a step towards a proof of this
statement.

######  Lemma 6.11.

If $f:D\to\R$, $D\subseteq\R$, $c$ is an accumulation point of $D$, and $f$ is
locally bounded at $c$, then the accretion $A(f,c,D\setminus\\{c\\})$ is a
nonempty compact set.

###### Proof.

Assume all of the hypotheses hold. By Lemma 6.5, $A(f,c,D\setminus\\{c\\})$ is
closed. Since $f$ is locally bounded at $c$, $A(f,c,D\setminus\\{c\\})$ is
bounded. So by the Heine-Borel Theorem [Abbott, Theorem 3.3.8, p.98],
$A(f,c,D\setminus\\{c\\})$ is compact.

Let $(x_{n})$ be a sequeunce of points in $D\setminus\\{c\\}$ where
$\lim_{x_{n}}=c$. It follows that $(f(x_{n}))$ is a bounded set of real
numbers. So by the Bolzano-Weierstrass Theorem [Abbott, Theorem 2.5.5, p.64],
there is a real number $y$ and a subsequence $(x_{n_{k}})$ of $(x_{n})$ such
that $\lim_{k\to\infty}f(x_{n_{k}})=y$. Since $\lim_{k\to\infty}x_{n_{k}}=c$
(see [Abbott, Theorem 2.5.2, p.63]), it follows that for every $\delta>0$
there is a positive integer $K(\delta)$ such that for all $k\geq K(\delta)$ it
must be that $x_{n_{k}}\in V_{\delta}(c)\cap D\setminus\\{c\\}$. Since
$\lim_{k\to\infty}f(x_{n_{k}})=y$ implies $y\in A((f(x_{n_{k}})))$ by
Proposition 5.7, it follows that $y\operatorname{acl}f(V_{\delta}(c)\cap
D\setminus\\{c\\})$ for every $\delta>0$. Therefore, $y\in
A(f,c,D\setminus\\{c\\})$. ∎

The following result, which is the last one in this section, connects various
equivalent statements regarding functional limits.

######  Theorem 6.12.

Let $D\subseteq\R$, let $f:D\to\R$, and let $c$ be an accumulation point of
$D$. Then the following statements are equivalent:

  * (i)

$\lim_{x\to c}f(x)=\ell$.

  * (ii)

For any sequence $(x_{n})$ of points in $D$ where $\lim_{n\to\infty}x_{n}=c$
and $x_{n}\neq c$ for all $n\in\mathbb{N}$, it follows that
$\lim_{n\to\infty}f(x_{n})=\ell$.

  * (iii)

$\operatorname*{alim}_{x\to c}f(x)=\ell$.

###### Proof.

The equivalence of (i) and (ii) is well known. See [Abbott, Theorem 4.2.3,
p.118], for instance. Thus, it suffices to show (ii) and (iii) are equivalent.

Throughout this proof, fix a real number $\ell$.

To show (ii) implies (iii), first suppose $f$ is not locally bounded at $c$.
Then there is a sequence $(z_{n})$ of points in $D\setminus\\{c\\}$ such that
$\lim_{n\to\infty}z_{n}=c$ but $|f(z_{n})|>\ell+1$ for large enough $n$. Then
$\lim_{n\to\infty}f(z_{n})\neq\ell$.

Now suppose $f$ is locally bounded at $c$ but
$A(f,c,D\setminus\\{c\\})\neq\\{\ell\\}$. By Lemma 6.11,
$A(f,c,D\setminus\\{c\\})$ is not empty, so there is some $y\in
A(f,c,D\setminus\\{c\\})$ where $\ell\neq y$. As in the proof of Lemma 6.11,
there is a sequence $(y_{k})$ of points in $D\setminus\\{c\\}$ such that
$\lim_{k\to\infty}y_{k}=c$ and $\lim_{k\to\infty}f(y_{k})=y\neq\ell$.

To show (iii) implies (ii), suppose there is a sequence $(x_{n})$ of points in
$D\setminus\\{c\\}$ such that $\lim_{n\to\infty}x_{n}=c$ but
$\lim_{n\to\infty}f(x_{n})\neq\ell$. If $(f(x_{n}))$ is unbounded, then $f$ is
not locally bounded at $c$. Therefore $\operatorname*{alim}_{x\to
c}f(x)\neq\ell$.

Now suppose $(f(x_{n}))$ is bounded and there is a sequence $(x_{n})$ of
points in $D\setminus\\{c\\}$ such that $\lim_{n\to\infty}x_{n}=c$ but
$\lim_{n\to\infty}f(x_{n})\neq\ell$. By the Bolzano-Weierstrass Theorem
[Abbott, Theorem 2.5.5, p.64], there is a real number $y$ and a subsequence
$(x_{n_{k}})$ of $(x_{n})$ such that $\lim_{k\to\infty}f(x_{n_{k}})=y$ where
$y\neq\ell$. Then for every $\delta>0$ there is a positive integer $K(\delta)$
such that for all $k\geq K(\delta)$ it follows that $x_{n_{k}}\in
V_{\delta}(c)\cap D\setminus\\{c\\}$. Hence,
$y\operatorname{acl}f(V_{\delta}(c)\cap D\setminus\\{c\\})$ for every
$\delta>0$. Therefore, $y\in A(f,c,D\setminus\\{c\\})$ and so
$A(f,c,D\setminus\\{c\\})\neq\\{\ell\\}$. Thus $\operatorname*{alim}_{x\to
c}f(x)\neq\ell$. ∎

The next section connects the notions of arbitrarily close and accretion of
functions to the classical notions of continuity and differentiation.

##  7\. ACCRETION, CONTINUITY, AND DIFFERENTIATION

As in the second half of the previous section, the examples and results
obtained in this remainder of this section are limited to functions to and
from the real line $\R$ with its standard metric for the sake of exposition.
The results obtained so far readily yield some connections to continuity,
derivatives, and integrals of real valued functions on the real line.

First, thanks to Theorem 6.12 and the deep classical connection between
functional limits and pointwise continuity, the next result readily follows.
Note that $c$ is assumed to be both an accumulation point and an element of
the domain.

######  Theorem 7.1.

Let $D\subseteq\R$, let $f:D\to\R$, and let $c$ be in an accumulation point of
$D$ that is also in $D$. Then the following statements are equivalent:

  * (i)

$\lim_{x\to c}f(x)=f(c)$.

  * (ii)

For any sequence $(x_{n})$ of points in $D$ where $\lim_{n\to\infty}x_{n}=c$
and $x_{n}\neq c$ for all $n\in\mathbb{N}$, it follows that
$\lim_{n\to\infty}f(x_{n})=f(c)$.

  * (iii)

$\operatorname*{alim}_{x\to c}f(x)=f(c)$.

###### Proof.

Statements (i) and (ii) are equivalent to the classical definition of
pointwise continuity for real valued function on the real line as found in
[Abbott, Definition 4.3.1, p.122] and proven with [Abbott, Theorem 4.3.2,
p.123]. The equivalence of statement (iii) with both (i) and (ii) follows from
Theorem 6.12. ∎

The equivalence of statement (iii) with classical notions of continuity in a
mildly weakened setting motivate the following definition.

######  Definition 7.2.

Let $D\subseteq\R$, let $f:D\to\R$, and let $c$ be an accumulation point of
$D$ that is also in $D$. Then $f$ is said to be accrete-continuous at $c$ if

(7.1) $\displaystyle f(c)=\operatorname*{alim}\limits_{x\to c}f(x).$

The next corollary follows readily.

######  Corollary 7.3.

Let $D\subseteq\R$, let $f:D\to\R$, and let $c$ be an accumulation point of
$D$ that is also in $D$. Then $f$ is accrete-continuous at $c$ if and only if
$f$ is locally bounded at $c$ and $A(f,c,D)=\\{f(c)\\}$.

###### Proof.

The result follows immediately from Definitions 6.10 and 7.2. ∎

Thus a real valued function on a set of real numbers is continuous at $c$ if
and only if $f$ is locally bounded at $c$ and the accretion of $f$ at $c$ with
respect to the domain is the singleton comprising $f(c)$. So, following
intuition, as $\delta>0$ is taken to be smaller and smaller the corresponding
outputs of $f$ become closer and closer to $f(c)$ and no other value.

######  Remark 7.4.

Corollary 7.3 tells us that Dirichlet’s function from Example 6.6 is not
continous anywhere on $\R$ since its accretion at any point contains two
points. Similarly, Thomae’s function from Example 6.7 is continuous at each
irrational number (where the accretion contains only 0) and discontinuous at
each rational number (where the accretion contains two points).

To see how the notion of accretion limit ties into differentiation, recall the
definition of derivative for real valued functions on the real line.

######  Definition 7.5.

Let $I\subseteq\R$ be an interval, let $g:I\to\R$, and let $c\in I$. Then the
derivative of $g$ at $c$ is defined by

(7.2) $\displaystyle g^{\prime}(c)=\lim_{x\to c}\frac{g(x)-g(c)}{x-c},$

provided this limit exists. In this case, $g$ is said to be differentiable at
$c$.

Thus, if a real valued function $g$ defined on an interval $I$ has a
derivative at $c$, then by Theorem 6.12 and Definition 6.10 it follows that
$g^{\prime}(c)$ is the only real number arbitrarily close to the set of
difference quotients that pass through the point $(c,g(c))$ restricted to the
deleted $\delta$-neighborhhood of $c$ for any given $\delta>0$.

The notation and terminology just above illustrates a notion often heard in
calculus when discussing derivatives: as $x$ gets closer and closer to $c$
(hence $\delta$ is taken to be smaller and smaller), the slopes of the secant
lines get closer and closer to the slope of the tangent line. Also, the
following definition is now motivated.

######  Definition 7.6.

Let $I\subseteq\R$ be an interval, let $g:I\to\R$, and let $c\in I$. The
function $g$ is said to be accrete-differentiable at $c$ if

(7.3) $\displaystyle g^{a}(c)=\operatorname*{alim}\limits_{x\to
c}\frac{g(x)-g(c)}{x-c},$

provided this accretion limit exists. In this case, $g^{a}(c)$ is called the
accretion derivative of $g$ at $c$.

When the accretion derivative $g^{a}(c)$ exists, it follows that

(7.4)
$\displaystyle\\{g^{a}(c)\\}=\bigcap_{\delta>0}\overline{\left\\{\frac{g(x)-g(c)}{x-c}:x\in
V_{\delta}(c)\cap(I\setminus\\{c\\})\right\\}}=A(q,c,I\setminus\\{c\\})$

where $q(x)$ is the difference quotient of $g$ at $c$ is defined for all $x$
in $I\setminus\\{c\\}$ by

(7.5) $\displaystyle q(x)=\frac{g(x)-g(c)}{x-c}.$

Also, a function is differentiable at a point if and only if it is accretion-
differentiable at the point.

######  Corollary 7.7.

Let $I\subseteq\R$ be an interval, let $g:I\to\R$, and let $c\in I$. Then $g$
is differentiable at $c$ if and only if $g$ is accrete-differentiable at $c$.

###### Proof.

The result follows immediately from Theorem 6.12 along with Definitions 7.5
and 7.6. ∎

An version of the Interior Extremum Theorem follows immediately from the
Definition 7.6 and the idea that 0 is the only real number arbitrarily close
to the sets of positive and negative numbers. See [Abbott, Theorem 5.2.6,
p.151] for comparison.

######  Theorem 7.8.

Suppose $f:(a,b)\to\R$, $f$ is accrete-differentiable at every point in the
open interval $(a,b)$, and $f$ attains its maximum at some point $c$ in
$(a,b)$. Then $f^{a}(c)=0$.

###### Proof.

Let $\delta>0$. Suppose $x,y\in(a,b)$ with $c-\delta<x<c<y<c+\delta$. Then

(7.6) $\displaystyle x-c<0<y-c,\quad f(x)-f(c)\leq 0,\quad\text{ and }\quad
f(y)-f(c)\leq 0.$

Hence

(7.7) $\displaystyle\frac{f(y)-f(c)}{y-c}\leq 0\leq\frac{f(x)-f(c)}{x-c}.$

Therefore, $f^{a}(c)$ is arbitrarily close to both the set of nonnegative real
numbers and the set of nonpositive real numbers. By Lemma 2.6, $f^{a}(c)=0$. ∎

##  8\. INTEGRATION AND THE EVALUATION PART OF THE FUNDAMENTAL THEOREM OF
CALCULUS

Following the content in Section 2, a bit more notation and terminology
actually provides enough material with to define an integral of a bounded real
valued function $f$ on a closed and bounded interval $[a,b]$.

######  Definition 8.1.

A partition of a closed and bounded interval $[a,b]$ is a finite set $P$ of
points of $[a,b]$ such that

(8.1) $\displaystyle
P=\\{x_{0}=a,x_{1},\ldots,x_{n}=b\\}\subseteq\R^{n}\quad\textnormal{where}\quad
x_{0}<x_{1}<\ldots<x_{n}.$

A partition of $[a,b]$ is also said to partition $[a,b]$. A weight is a vector
$\textbf{w}\in\bigcup_{m\in\mathbb{N}}\R^{m}$. For a partition of $[a,b]$
given by $P=\\{x_{0},\ldots,x_{n}\\}$ and any weight
$\textbf{v}=(v_{1},\ldots,v_{n})$ in $\R^{n}$, the weighted sum
$s(P,\textbf{v})$ is defined by

(8.2) $\displaystyle s(P,\textbf{v})=\sum_{k=1}^{n}v_{k}(x_{k}-x_{k-1}).$

Weighted sums allow one to literally think outside the box. Given a bounded
function on a compact interval, the endpoints of this compact interval along
with the supremum and infimum of the range of the function form a box in which
the graph of the function resides. The idea is to approximate the integral in
question by choosing values to represent the height of the function which lie
outside of this box, then refining the approximation with finer and finer
partitions, thus finer and finer boxes. The result is an alternative
definition of integration that relies on neither Riemann sums nor Darboux
sums.

######  Definition 8.2.

Let $f:[a,b]\rightarrow\R$ and suppose $f$ is bounded. Let $P$ be a partition
of $[a,b]$ given by $P=\\{x_{o},\ldots,x_{n}\\}$ with
$x_{0}<x_{1}<\ldots<x_{n}$. The set of upper weights of $f$ with respect to
$P$ is defined by

(8.3) $\displaystyle
U(f,P)=\\{\textbf{a}=(a_{1},\ldots,a_{n})\in\R^{n}:a_{k}\geq f(x)\text{ for
}x\in[x_{k-1},x_{k}],\,k=1,\ldots,n\\}.$

Similarly, the set of lower weights of $f$ with respect to $P$ is defined by

(8.4) $\displaystyle
L(f,P)=\\{\textbf{b}=(b_{1},\ldots,b_{n})\in\R^{n}:b_{k}\leq f(x)\text{ for
}x\in[x_{k-1},x_{k}],\,k=1,\ldots,n\\}.$

Lastly, the set of upper sums of $f$ is defined by

(8.5) $\displaystyle U_{f}=\\{s(P,\textbf{a}):P\text{ is a partition of
}[a,b],\textbf{a}\in U(f,P)\\},$

and the set of lower sums of $f$ is defined by

(8.6) $\displaystyle L_{f}=\\{s(P,\textbf{a}):P\text{ is a partition of
}[a,b],\textbf{b}\in L(f,P)\\}.$

Note that $U_{f}$ and $L_{f}$ are sets of real numbers while $U(f,P)$ and
$L(f,P)$ are sets of vectors.

In Definition 8.3, real number arbitrarily close to both $L_{f}$ and $U_{f}$
is the integral of $f$. Moreover, the integral defined in this way makes use
of neither tags nor suprema and infima. Thus, Definition 8.3 is distinct from
the Darboux and Riemann integrals. Still, these notions of integration are
equivalent, as shown in Theorem 8.8.

######  Definition 8.3.

A bounded function $f:[a,b]\to\R$ is said to be integrable over $[a,b]$ if
there is some $y\in\R$ such that $y$ is arbitrarily close to both $U_{f}$ and
$L_{f}$. In this case, $y$ is denoted by $\int_{a}^{b}f$ and called the
integral of $f$ over $[a,b]$.

After developing a couple of basic results in this framework, a version of the
evaluation part of the Fundamental Theorem of Calculus can be proven. (See
Theorem 8.7 below.)

First, the following lemma establishes the fact that any upper sum of a given
function is greater than or equal to any lower sum of the same function,
regardless of the choice of partitions.

######  Lemma 8.4.

For a bounded function $f:[a,b]\to\R$, any pair of partitions $P_{1}$ and
$P_{2}$ of $[a,b]$, and any choice of an upper weight $\textnormal{{a}}\in
U(f,P_{1})$ and a lower weight $\textnormal{{b}}\in L(f,P_{2})$, it follows
that $s(P_{2},\textnormal{{b}})\leq s(P_{1},\textnormal{{a}})$.

###### Proof.

Let $P_{1}=\\{z_{0},\ldots,z_{n_{1}}\\}$ and
$P_{2}=\\{y_{0},\ldots,y_{n_{2}}\\}$. Define $P=P_{1}\cup P_{2}$ and rewrite
$P$ as

(8.7) $\displaystyle P=\\{x_{0},x_{1},\ldots,x_{n}\\}\text{ such that
}x_{0}<x_{1}<\ldots<x_{n}.$

Given $\textbf{a}=(a_{1},\ldots,a_{n_{1}})\in U(f,P_{1})$ and
$\textbf{b}=(b_{1},\dots,a_{n_{2}})\in L(f,P_{2}),$ define

(8.8) $\displaystyle\textbf{a}^{\prime}$
$\displaystyle=(a_{1}^{\prime},\ldots,a_{n}^{\prime})\in U(f,P)$

by $a^{\prime}_{k}=a_{h}$ when $[x_{k-1},x_{k}]\subseteq[z_{h-1},z_{h}]$ for
$k=1,\ldots,n$ and $h=1,\ldots,n_{1}$, accordingly. Likewise, define

(8.9)
$\displaystyle\textbf{b}^{\prime}=(b_{1}^{\prime},\ldots,b_{n}^{\prime})\in
L(f,P)$

by $b^{\prime}_{k}=b_{j}$ when $[x_{k-1},x_{k}]\subseteq[y_{j-1},y_{j}]$ for
$k=1,\ldots,n$ and $j=1,\ldots,n_{2}$, accordingly. Since $b^{\prime}_{k}\leq
f(x)\leq a^{\prime}_{k}$ for each $k=1,\ldots,n$ and any
$x\in[x_{k-1},x_{k}]$, partial telescoping yields

(8.10) $\displaystyle
s(P_{2},\textbf{b})=s(P,\textbf{b}^{\prime})\leq\sum_{k=1}^{n}f(x)(x_{k}-x_{k-1})\leq
s(P,\textbf{a}^{\prime})=s(P_{1},\textbf{a}).$

∎

Lemma 8.4 leads immediately to the following result.

######  Lemma 8.5.

The integral of $f$ over $[a,b]$ is unique.

###### Proof.

By way of contradiction, suppose $y_{2}<y_{1}$ but
$y_{1}\operatorname{acl}U_{f},y_{1}\operatorname{acl}L_{f},y_{2}\operatorname{acl}U_{f}$,
and $y_{2}\operatorname{acl}L_{f}$. Choose
$\varepsilon_{0}=|y_{1}-y_{2}|/3>0$. Then there are partitions $P_{1}$ and
$P_{2}$ of $[a,b]$ along with an upper weight $\textbf{a}\in U(f,P_{1})$ and a
lower weight $\textbf{b}\in L(f,P_{2})$ such that

(8.11)
$\displaystyle|s(P_{1},\textbf{a})-y_{2}|<\varepsilon_{0}\qquad\textnormal{and}\qquad|s(P_{2},\textbf{b})-y_{1}|<\varepsilon_{0}.$

Hence $-\varepsilon_{0}<s(P_{1},\textbf{a})-y_{2}<\varepsilon_{0}$ and
$-\varepsilon_{0}<s(P_{2},\textbf{b})-y_{1}<\varepsilon_{0}$. Since

(8.12) $\displaystyle
y_{2}+\varepsilon_{0}=\frac{2y_{2}+y_{1}}{3}<\frac{y_{2}+2y_{1}}{3}=y_{1}-\varepsilon_{0},$

it follows that

(8.13) $\displaystyle
s(P_{1},\textbf{a})<y_{2}+\varepsilon_{0}<y_{1}-\varepsilon_{0}<s(P_{2},\textbf{b}).$

These inequalities contradict Lemma 8.4 which says $s(P_{2},\textbf{b})\leq
s(P_{1},\textbf{a}).$ Therefore, $\int_{a}^{b}f$ is unique. ∎

The Mean Value Theorem provides a key piece of the puzzle. Its statement is
provided here for convenience, but see [Abbott, Theorem 5.3.2, p.156] for a
proof.

######  Theorem 8.6 (Mean Value Theorem).

If $f:[a,b]\to\R$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then
there is a point $c\in(a,b)$ where

(8.14) $\displaystyle f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}.$

A version of the evaluation half of the Fundamental Theorem of Calculus now
follows readily from the definition of an integral given in Definition 8.3 and
the results in this section. In particular, neither Riemann sums nor Darboux
sums are used in the proof.

######  Theorem 8.7 (Evaluation half of the Fundamental Theorem of Calculus).

Suppose $f$ and $F$ are real valued functions on $[a,b]$ where $f$ is the
derivative of $F$ on $[a,b]$ and $f$ is integrable over $[a,b]$. Then
$\int_{a}^{b}f=F(b)-F(a)$.

###### Proof.

Let $P$ be a partition of $[a,b]$ with $P=\\{x_{0},\ldots,x_{n}\\}$. By the
Mean Value Theorem 8.6, for each $k=1,\ldots,n$ there exists
$t_{k}\in(x_{k-1},x_{k})$ such that

(8.15) $\displaystyle
F(x_{k})-F(x_{k-1})=F^{\prime}(t_{k})(x_{k}-x_{k-1})=f(t_{k})(x_{k}-x_{k-1}).$

Since $f$ is integrable, it is bounded on $[a,b]$. By Lemma 8.4, any choice of
weights $\textbf{a}\in U(f,P)$ and $\textbf{b}\in L(f,P)$ as well as any index
$k=1,\ldots,n$ yields

(8.16) $\displaystyle b_{k}\leq f(t_{k})\leq a_{k}\quad\text{ and }\quad
x_{k}-x_{k-1}>0.$

Therefore,

(8.17) $\displaystyle
s(P,\textbf{b})\leq\sum_{k=1}^{n}f(t_{k})(x_{k}-x_{k-1})=F(b)-F(a)\leq
s(P,\textbf{a}).$

Now, since $f$ is integrable over $[a,b]$, for every $\varepsilon>0$ there are
partitions $P_{1}$ and $P_{2}$ of $[a,b]$ along with an upper weight
$\textbf{a}_{1}\in U(f,P_{1})$ and a lower weight $\textbf{b}_{2}\in
L(f,P_{2})$ such that

(8.18)
$\displaystyle\left|\int_{a}^{b}f-s(P_{1},\textbf{a}_{1})\right|<\varepsilon\quad\textnormal{and}\quad\left|\int_{a}^{b}f-s(P_{2},\textbf{b}_{2})\right|<\varepsilon.$

By Lemma 8.4 and inequalities (8.18),

(8.19) $\displaystyle s(P_{2},\textbf{b}_{2})-\varepsilon$ $\displaystyle\leq
F(b)-F(a)-\varepsilon\leq s(P_{1},\textbf{a}_{1})-\varepsilon<\int_{a}^{b}f$

and

(8.20) $\displaystyle\int_{a}^{b}f$
$\displaystyle<s(P_{2},\textbf{b}_{2})+\varepsilon\leq
F(b)-F(a)+\varepsilon\leq s(P_{1},\textbf{a}_{1})+\varepsilon.$

Therefore, for every $\varepsilon>0$ it follows that

(8.21) $\displaystyle\left|\int_{a}^{b}f-(F(b)-F(a))\right|<\varepsilon.$

By Lemma 2.5,

(8.22) $\displaystyle\int_{a}^{b}f=F(b)-F(a).$

∎

The proof of this version of the evaluation half of the Fundamental Theorem of
Calculus (Theorem 8.7) used neither the tags associated Riemann sums and nor
the suprema and infima associated with Darboux sums. Still, the three
corresponding notions of integration are equivalent.

######  Theorem 8.8.

Let $f$ be a bounded real valued function on $[a,b]$. The following are
equivalent$:$

  * (i)

$f$ is integrable over $[a,b]$ as defined in Definition 8.3.

  * (ii)

$f$ is Darboux-integrable over $[a,b]$ ([Abbott, Definition 7.2.7, p.220]).

  * (iii)

$f$ is Riemann-integrable over $[a,b]$ ([Abbott, Theorem 8.1.2, p.251]).

The equivalence of (ii) and (iii) is a classical result in analysis. See
[Abbott, Theorem 8.1.2, p.251] for details. To explore the possible
equivalence of (i) with (ii) and (iii), but in lieu of fully developing
Darboux and Riemann integrals on their own, consider the various sums used in
each of the definitions of integrability, respectively.

To establish the connection to Riemann sums, let $P=\\{x_{0},\ldots,x_{n}\\}$
be a partition of $[a,b]$. Consider a choice of a vector
$\textbf{c}=(c_{1},\ldots,c_{n})\in\R^{n}$ where $c_{k}\in[x_{k-1},x_{k}]$ for
each $k=1,\ldots,n$, define

(8.23) $\displaystyle\textbf{f}(\textbf{c})=(f(c_{1}),\ldots,f(c_{n})).$

Then the classical Riemann sum (see [Abbott, p.250]) with ‘tags’
$\textbf{c}=(c_{1},\ldots,c_{n})\in\R^{n}$ is given by the weighted sum with
weights given by $\textbf{f}(\textbf{c})$ as follows:

(8.24) $\displaystyle
s(P,\textbf{f}(\textbf{c}))=\sum_{k=1}^{n}f(c_{k})(x_{k}-x_{k-1}).$

As a result, for any partition $P$ of $[a,b]$ and any choice of upper weights
$\textbf{a}\in U(f,P)$ and lower weights $\textbf{b}\in L(f,P)$, it follows
that since $b_{k}\leq f(x)\leq a_{k}$ for each $k=1,\ldots,n$ and any
$x\in[x_{k-1},x_{k}]$

(8.25) $\displaystyle
s(P_{2},\textbf{b})\leq\sum_{k=1}^{n}f(x)(x_{k}-x_{k-1})\leq
s(P_{1},\textbf{a}).$

Now, to establish the connection to Darboux sums, again let
$P=\\{x_{0},\ldots,x_{n}\\}$ be a partition of $[a,b]$. Define
$\textbf{u}=(u_{1},\ldots,u_{n})\in U(f,P)$ by

(8.26) $\displaystyle u_{k}=\sup\\{f(x):x\in[x_{k-1},x_{k}]\\}$

and define $\textbf{w}=(w_{1},\ldots,w_{n})\in L(f,P)$ by

(8.27) $\displaystyle w_{k}=\inf\\{f(x):x\in[x_{k-1},x_{k}]\\},$

Then the classical upper and lower Darboux sums (see [Abbott, Definition
7.2.1, p.218]) are given by the weighted sums weighted by u and w,
respectively. That is,

(8.28) $\displaystyle
s(P,\textbf{u})=\sum_{k=1}^{n}u_{k}(x_{k}-x_{k-1})\qquad\textnormal{and}\qquad
s(P,\textbf{w})=\sum_{k=1}^{n}w_{k}(x_{k}-x_{k-1}).$

So for any $\textbf{a}\in U(f,P)$ and $\textbf{b}\in L(f,P)$ it follows that

(8.29) $\displaystyle s(P,\textbf{b})\leq s(P,\textbf{w})\leq
s(P,\textbf{f}(\textbf{c}))\leq s(P,\textbf{u})\leq s(P,\textbf{a}).$

To complete the proof of Theorem 8.8 it suffice to show (i) is equivalent to
(ii). To this end, the following equivalent form of the Darboux integral
[Abbott, Theorem 7.2.8, p.221]—which itself is reminiscent of the definition
of arbitrarily close—is quite useful.

######  Theorem 8.9.

A bounded function $f$ is integrable on $[a,b]$ if and only if, for every
$\varepsilon>0$, there exists a partition $P_{\varepsilon}$ of $[a,b]$ such
that

(8.30) $\displaystyle
s(P,\text{{\em$\textbf{u}$}})-s(P,\text{{\em$\textbf{w}$}})<\varepsilon,$

where u and w are as in (8.26) and (8.27), respectively.

###### Proof of Theorem 8.8.

Assume $f$ is Darboux-integrable over $[a,b]$. Let $\int_{a}^{b}f$ denote the
Darboux integral of $f$, which by [Abbott, Definition 7.2.7, p.220] yields

(8.31) $\displaystyle\int_{a}^{b}f=\inf\\{s(P,\textbf{u}):\textnormal{upper
Darboux sum }s(P,\textbf{u}),P\textnormal{ partitions }[a,b]\\}$

and

(8.32) $\displaystyle\int_{a}^{b}f=\sup\\{s(P,\textbf{w}):\textnormal{lower
Darboux sum }s(P,\textbf{w}),P\textnormal{ partitions }[a,b]\\}.$

Since $s(P,\textbf{u})\in U_{f}$ and $s(P,\textbf{w})\in L_{f}$ for any
corresponding weights u and w defined as in (8.26) and (8.27), respectively,
it follows from Definition 3.2 that $\int_{a}^{b}f\operatorname{acl}U_{f}$ and
$\int_{a}^{b}f\operatorname{acl}L_{f}$. Therefore, $f$ is integrable over
$[a,b]$ in the sense of Definition 8.3.

Now suppose $f$ is integrable over $[a,b]$ in the sense of Definition 8.3
where $\int_{a}^{b}f\operatorname{acl}U_{f}$ and
$\int_{a}^{b}f\operatorname{acl}L_{f}$. Let $\varepsilon>0$. Then there is a
partition $P$ of $[a,b]$ along with an upper weight $\textbf{a}\in U(f,P)$ and
a lower weight $\textbf{b}\in L(f,P)$ such that

(8.33)
$\displaystyle\left|\int_{a}^{b}f-s(P,\textbf{a})\right|<\varepsilon/2\qquad\text{and}\qquad\left|\int_{a}^{b}f-s(P,\textbf{b})\right|<\varepsilon/2.$

Let weights u and w be defined as in (8.26) and (8.27), respectively. Then

(8.34) $\displaystyle s(P,\textbf{b})\leq s(P,\textbf{w})\leq
s(P,\textbf{u})\leq s(P,\textbf{a})$

from which it follows that

(8.35)
$\displaystyle-s(P,\textbf{a})\leq-s(P,\textbf{u})\leq-s(P,\textbf{w})\leq-s(P,\textbf{b}).$

A combination of inequalities (8.33), (8.34), and (8.35) yields

(8.36) $\displaystyle-\varepsilon/2<s(P,\textbf{b})-\int_{a}^{b}f\leq
s(P,\textbf{u})-\int_{a}^{b}f\leq s(P,\textbf{a})-\int_{a}^{b}f<\varepsilon/2$

and

(8.37)
$\displaystyle\varepsilon/2<\int_{a}^{b}f-s(P,\textbf{a})\leq\int_{a}^{b}f-s(P,\textbf{w})\leq\int_{a}^{b}f-s(P,\textbf{b})<\varepsilon/2.$

Therefore,

(8.38) $\displaystyle s(P,\textbf{u})-s(P,\textbf{w})<\varepsilon.$

By Theorem 8.9, $f$ is Darboux-integrable over $[a,b]$. ∎

The paper closes with some thoughts on pedagogy and some exercises.

##  9\. CLOSURE: PEDAGOGY AND EXERCISES

A quantifier-heavy way to write out the definition of arbitrarily close,
Definition 2.2, in the context of the real line is as follows:

(9.1)
$\displaystyle\ell\operatorname{acl}S\quad\Longleftrightarrow\quad\forall\,\varepsilon>0,\exists\,x\in
S\textnormal{ such that }|x-\ell|<\varepsilon.$

For the sake of comparison, consider the definition of the limit of a sequence
(Definition 2.8) stated with quantifiers:

(9.2)
$\displaystyle\lim_{n\to\infty}a_{n}=\ell\quad\Longleftrightarrow\quad\forall\,\varepsilon>0,\exists\,N\in\mathbb{N}\textnormal{
such that }\forall\,n\geq N,|a_{n}-\ell|<\varepsilon.$

Statement (9.2) is difficult for students to understand at first, which is why
one should consider using a technical definition of arbitrarily close such as
Definition 2.2 as a bridge to limits and convergence.

Note that only two qualifiers are used in the definition of arbitrarily close.
For this reason, discussions of arbitrarily close could reasonably be part of
any course that serves as an introduction to writing mathematical proofs.

Actually, there are topics in undergraduate real analysis that could
reasonably be included in a proofs-based course. For instance, writing logical
statements in the manner just above is not always the best way to understand
them, but doing so allows for negations to be readily stated. In the case
where a real number $y$ is not arbitrarily close to a set of real number $B$,
one could write

(9.3) $\displaystyle\exists\,\varepsilon_{0}>0\textnormal{ such that
}\forall\,x\in B\textnormal{ it follows that }|x-y|\geq\varepsilon_{0}.$

Thus, the negation of arbitrarily close leads to a statement that is
essentially the property that defines what it means for a set to be open in
the topology of the real line, as discussed in the dialog leading to
Definition 4.3. Look at the statement again. It immediately implies that each
real number $y$ that is not arbitrarily close to $B$ lies in an open interval
that does not intersect $B$.

Also consider the case when a set has an upper bound. Such a set is said to be
bounded above. See Definition 3.1 above as well as [Abbott, Definition 1.3.1,
p.15]. The intuition students have by the time they reach a first course in
analysis serves them well when it comes to dealing with bounded and unbounded
sets. In terms of quantifiers, a set $S$ is said to be bounded when

(9.4) $\displaystyle\exists\,b>0\textnormal{ such that }\forall\,x\in
S\textnormal{ it follows that }|x|\leq b.$

Of course, if $S$ is unbounded, it follows that

(9.5) $\displaystyle\forall\,b>0,\exists\,x\in S\textnormal{ such that
}|x|>b.$

A simple intuitive idea is that quantified statements of the form
$\exists\ldots$ followed by $\forall\ldots$ seem to indicate the existence of
a constraint of some kind. That is, whatever exists dictates something about
everything in some collection of objects. On the other hand, statements of the
form $\forall\ldots$ followed by $\exists\ldots$ definitely indicate the
existence of a function. In particular, the first pairing of quantifiers in
the definition of the limit of sequence (Definition 2.8) given by

(9.6) $\displaystyle\forall\,\varepsilon>0,\exists\,N\in\mathbb{N}\ldots$

is often taken to mean that $N$ is a function of the input given by the error
$\varepsilon>0$. This can be a helpful perspective to take when teaching this
topic. Likewise, the second pairing

(9.7) $\displaystyle\exists\,N\in\mathbb{N}\textnormal{ such that
}\forall\,n\geq N\ldots$

can be taken to mean the $N$ provides a constraint on all of the indeces $n$
that follow. Therefore, $N$ is a function of $\varepsilon>0$ that gives a
constraint on $n$ which indicates how far into the sequence one needs to go to
ensure the terms are as close to the limit as desired.

Ultimately, an important question is as follows: Where should an introductory
course on real analysis begin? One might consider kicking things off with a
formal definition of something many mathematicians seem to intuitively
understand: What it means for things to be arbitrarily close. This can be done
even before the definition of supremum and certainly before any notion of
convergence or limit is discussed. The time spent on such a definition could
serve as an intermediate step towards an understanding of the subtleties of
convergence and other abstract mathematical concepts.

Many of the results proven throughout the paper would serve as an interesting
exercise in a course on undergraduate analysis. The paper concludes with
additional exercises for the reader to try. It is suggested that solutions to
these exercises be stated in terms of the technical definition of arbitrarily
close along with the various notions of accretion presented in this paper.

######  Exercise 9.1.

Let $a$ and $b$ be two points in a metric space. Prove $a=b$ if and only if
$a\operatorname{acl}\\{b\\}$.