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$\displaystyle\boldsymbol{a}\cdot\left(8R_{0}^{2}\,\boldsymbol{b}-2||\boldsymbol{b}-\boldsymbol{c}||_{2}^{2}\,\boldsymbol{c}\right)\leq\left|4R_{0}^{2}\cdot\left(R_{0}^{2}+||\boldsymbol{b}||_{2}^{2}\right)-||\boldsymbol{b}-\boldsymbol{c}||_{2}^{2}\cdot\left(R_{0}^{2}+||\boldsymbol{c}||_{2}^{2}\right)\right|.$

By utilizing the above bound, inequality (77) is proved by noting that its RHS
is non-negative, which can be proved using inequality (79). As mentioned
earlier, this implies inequality (76).

To proceed further, we note that $\boldsymbol{b}-\boldsymbol{c}$ lies in the
eigenspace of $H_{m^{*}}^{\prime}$ associated with eigenvalue $m^{*}$. Hence,

$\displaystyle
H_{m^{*}}^{\prime}\left(\boldsymbol{a}-\boldsymbol{c}\right)=H_{m^{*}}^{\prime}\left(\boldsymbol{a}-\boldsymbol{b}\right)+m^{*}\left(\boldsymbol{b}-\boldsymbol{c}\right).$

Therefore, by triangle inequality, we have

$\displaystyle\left|\left|H^{\prime}_{m^{*}}\left(H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right)\right|\right|_{2}\leq$
$\displaystyle\left|\left|H_{m^{*}}^{\prime}\left(\boldsymbol{a}-\boldsymbol{b}\right)\right|\right|_{2}+m^{*}\left|\left|\boldsymbol{b}-\boldsymbol{c}\right|\right|_{2}.$
(80)

Note that by inequality (76) and the fact that all eigenvalues of
$H_{m^{*}}^{\prime}$ are lower bounded by $m^{*}$, we have

$\displaystyle
m^{*}\left|\left|\boldsymbol{b}-\boldsymbol{c}\right|\right|_{2}\leq\frac{2R_{0}}{||\boldsymbol{a}-\boldsymbol{c}||_{2}}\left|\left|H_{m^{*}}^{\prime}\left(\boldsymbol{a}-\boldsymbol{b}\right)\right|\right|_{2}.$

Therefore, it remains to upper bound the $\ell_{2}$ norm of
$H_{m^{*}}^{\prime}\left(\boldsymbol{a}-\boldsymbol{b}\right)$.

By the definition of vectors $\boldsymbol{a}$, $\boldsymbol{b}$,

$\displaystyle H_{m^{*}}^{\prime}\left(\boldsymbol{a}-\boldsymbol{b}\right)$
$\displaystyle=(\boldsymbol{m}-\boldsymbol{m}^{\prime})+(H_{m^{*}}^{\prime}-H_{m^{*}})\,\boldsymbol{a}.$
(81)

The triangle inequality implies that

$\displaystyle\left|\left|H_{m^{*}}^{\prime}\left(\boldsymbol{a}-\boldsymbol{b}\right)\right|\right|_{2}$
$\displaystyle\leq\left|\left|\boldsymbol{m}-\boldsymbol{m}^{\prime}\right|\right|_{2}+R_{0}\left|\left|H_{m^{*}}^{\prime}-H_{m^{*}}\right|\right|_{\textup{F}}.$
(82)

Hence,

$\displaystyle\left|\left|H^{\prime}_{m^{*}}\left(H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right)\right|\right|_{2}\leq$
$\displaystyle\left(1+\frac{2R_{0}}{||\boldsymbol{a}-\boldsymbol{c}||_{2}}\right)$
$\displaystyle\cdot\left(\left|\left|\boldsymbol{m}-\boldsymbol{m}^{\prime}\right|\right|_{2}+R_{0}\left|\left|H_{m^{*}}^{\prime}-H_{m^{*}}\right|\right|_{\textup{F}}\right)$
$\displaystyle\leq$
$\displaystyle\left(1+\frac{2R_{0}}{||H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}||_{2}}\right)$
$\displaystyle\cdot\left(\left|\left|\boldsymbol{m}-\boldsymbol{m}^{\prime}\right|\right|_{2}+R_{0}\left|\left|H-H^{\prime}\right|\right|_{\textup{F}}\right),$

where the last step is due to inequality (71). Thus, inequality (E.4) is
implied by the semi-positive-definiteness of
$H^{\prime}_{m^{*}}-H^{\prime}_{m^{\prime*}}$.

Finally, when $m^{*}<m^{\prime*}$, we let $H_{m^{\prime*}}$ denote the
symmetric matrix sharing the same eigenbasis of $H$, but with each eigenvalue
$\lambda$ replaced by $\max\\{\lambda,m^{\prime*}\\}$. Due to the equivalence
of $H$ and $H^{\prime}$, our earlier proof steps imply that

$\displaystyle\left|\left|H_{m*}\left(H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right)\right|\right|_{2}\leq$
$\displaystyle\left(1+\frac{2R_{0}}{||H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}||_{2}}\right)$
$\displaystyle\cdot\left(\left|\left|\boldsymbol{m}-\boldsymbol{m}^{\prime}\right|\right|_{2}+R_{0}\left|\left|H-H^{\prime}\right|\right|_{\textup{F}}\right).$

Hence, by triangle inequality, we can use the above bound as follows.

$\displaystyle\left|\left|H^{\prime}_{m^{\prime*}}\left(H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right)\right|\right|_{2}\leq$
$\displaystyle\left|\left|\left(H_{m^{\prime*}}-H^{\prime}_{m^{\prime*}}\right)\left(H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right)\right|\right|_{2}$
$\displaystyle+\left|\left|H_{m^{\prime*}}\left(H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right)\right|\right|_{2}$
$\displaystyle\leq$
$\displaystyle\left|\left|H_{m^{\prime*}}-H^{\prime}_{m^{\prime*}}\right|\right|_{\textup{F}}\left|\left|H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right|\right|_{2}$
$\displaystyle+\left|\left|H_{m^{\prime*}}\left(H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right)\right|\right|_{2}.$

Note that
$\left|\left|H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right|\right|_{2}\leq
2R_{0}$. By inequality (71), it is clear that

$\displaystyle\left|\left|H^{\prime}_{m^{\prime*}}\left(H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}\right)\right|\right|_{2}\leq$
$\displaystyle\left(3+\frac{2R_{0}}{||H_{m^{*}}^{-1}\boldsymbol{m}-H_{m^{\prime*}}^{\prime-1}\boldsymbol{m}^{\prime}||_{2}}\right)$
$\displaystyle\cdot\left(\left|\left|\boldsymbol{m}-\boldsymbol{m}^{\prime}\right|\right|_{2}+R_{0}\left|\left|H-H^{\prime}\right|\right|_{\textup{F}}\right).$

∎