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\begin{document}
\noindent
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\begin{center}
{{\Large\bf Section 3.1:} \bf A, 2}
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\begin{enumerate}

\item[A.] Let $(X,d)$ be a metric space and $A\subset X$ be a
  non-empty subset.  Recall that $d(x,A)\doteq\inf\{d(x,y):\, y\in
  A\}$.  Theorem 11.1.3 asserts that
  \begin{align*}
    |d(x,A)-d(y,A)|\leq d(x,y)\mbox{ for all } x,y\in A~.
  \end{align*}
  Prove this result.
  
\item[2.] Suppose that $(X,\|\cdot\|)$ is a uniformly convex Banach
  space, and let $Z\subset X$ be a non-empty, closed, convex subset.
  Let $P:X\to Z$ be the projection operator defined in Theorem 3.1.7.
  Show that $P$ is continuous. 
  \textbf{Hint:} Let $(x_n)_{n\in\NN}$ be a sequence converging to
  $x\in X$.  Argue that $\limsup \|x-P(x_n)\|\leq \|x-P(x)\|$ and
  $\liminf \|x-P(x_n)\|\geq \|x-P(x)\|$, and deduce from this that
  $\lim \|x-P(x_n)\|=\|x-P(x)\|$.  You may then ``recycle'' (or cite)
  arguments from the proof of Theorem 3.1.7 to ultimately argue that
  $\lim\|P(x_n)-P(x)\|=0$.
  
\end{enumerate}


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