Bounded Sequences in a Separable Real Hilbert Space Have Weakly Convergent Subsequences

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, let $\mathbb{N}$ be the set of natural numbers, and let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and distance $d$ such that $(H,d)$ is separable. Let $C\in\mathbb{R}$ and let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $H$ with $|x_{m}|\le C$ for every $m\in\mathbb{N}$.

Then there exist a subsequence $(x_{n_{j}})_{j\in\mathbb{N}}$ of $(x_{m})_{m\in\mathbb{N}}$ and a point $x\in H$ with $|x|\le C$ such that $(x_{n_{j}})_{j\in\mathbb{N}}$ converges weakly to $x$ in $H$.