On the uniqueness of certain weak cluster point in Hilbert space

On the uniqueness of certain weak cluster point in Hilbert space

The problem goes as follows: Let $\mathcal{H}$ be real Hilbert space,
$C\subset \mathcal{H}$ be a subset. Let $\{x_n\}\subset\mathcal{H}$
satisfies the following property: $$||x_{n+1}-x||\leq||x_n-x||$$ holds for
all $x \in C$. Show that $\{x_n\}$ has at most one weak cluster point in
$C$.
I tried to use Mazur's theorem to find some convex combination of
$\{x_n\}$ making the weak convergence into strong one, but then the
contraction property does not hold anyway. Though it is an easy
observation that $\{||x_n-x||\}$ is convergent sequence for any $\forall x
\in C$, I had no idea of whether this may be of help or not.