\noindent
%\subsection{Proof of Lemma \thelecture.6}
\textcolor{red}{{\bf PROOF OF LEMMA \thelecture.6}}
\noindent
\disp{x_{k_0+j} = x_{k_0+1} = x_*}
for all $j>0$, where $k_0$ is index of last successful iterate.
\vspace*{0,2in}
\noindent
All iterations are unsuccessful for sufficiently large $k$ \implies
$\{\Delta_k\} \longrightarrow 0$
\vspace*{0,2in}
\noindent + Lemma~\thelecture.4 then implies that if
$\|g_{k_0+1}\| > 0$
there must be a successful iteration of index larger than $k_0$, which is
impossible \implies $\|g_{k_0+1}\| =0$.