Convergence of distance in Hilbert spaces

Question

Let $X$ be a Hilbert space, $C$ be a closed subset of $X$ and {x_n} be a bounded sequence in X. Let $d(\cdot,C)$ be the distance function to $C$. Assume that $d(x_n,C) \to 0$ as $n\to \infty$. Can we claim that $x_n$ converges to some point in $C$?

Answer 1

In general, you only get weak convergence of a subsequence towards a point in $C$.

Think about the case $C = X$ or $C$ being a closed subspace for simple examples.

Answer 2

Note that $\vert C \vert \geq 2$ is the interesting case. Let $x,y \in C$ and $x\neq y$. Now construct $$x_n=\begin{cases} x, \quad \text{if } n \text{ is even} \\ y, \quad \text{else} \end{cases}.$$ Because $x,y \in C$ we have $d(x_n, C) = 0$ for all $n \in \mathbb N$. It is obvious that $(x_n)_{n \in \mathbb N}$ doesn't converges. Like gerw noted you can observe a convergent subsequence.

Since Hilbert spaces are reflexive you can deduce, that bounded sequences have a weak convergent subsequence.