Every bounded sequence in a real Hilbert space has a weakly convergent subsequence

Let $H$ be a real Hilbert space and let $(x_n)_{n\ge0}$ be a norm-bounded sequence: there is a real $R$ such that $\|x_n\|\le R$ for every $n$. Then there are a strictly increasing map $r:\mathbb N\to\mathbb N$ and a point $p\in H$ such that $x_{r(k)}\rightharpoonup p$. Here weak convergence means $\langle x_{r(k)},y\rangle\to\langle p,y\rangle$ for every $y\in H$. No separability or finite-dimensionality assumption is imposed on $H$. This is weak sequential compactness for bounded sequences in a Hilbert space, used as the compactness step in quasi-Fejer/Opial convergence arguments.