Showing that there exists a sequence that converges weakly in $H_0^1(\Omega)$.

Question

Proof of lemma $9.7$ in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations argues as follows:

For an element $u \in H_0^1(\Omega)$ we define $D_h u= \frac{u(x+h)-u(x)}{|h|}$, for $h \in \mathbb{R}^n$. Let's assume that $D_hu \in H_0^1(\Omega)$ and that: $$ ||D_hu||_{H^1(\Omega)} \leq || u||_{H^2(\Omega)}$$

Thus there exists a sequence $h_n \to 0$ such that $D_{h_n}u$ converges weakly to $g \in H_0^1(\Omega)$ (Since $H_0^1(\Omega)$ is a Hilbert Space).

Could someone explain me why do we have this result about weak convergence?

Answer 1

Any bounded sequence in a separable Hilbert space (which is reflexive) has a weakly convergent subsequence.

Added on edit: See Theorem 3.18 of the same book.