I'm trying to solve below exercise in Brezis' Functional Analysis
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $u_n, u \in H$ such that $u_n \to u$ in the weak topology $\sigma (H, H^*)$ and that $\limsup_n |u_n| \le |u|$. Then $u_n \to u$ in norm topology.
Could you verify my below attempt?
We have $|u_n-u|^2 = |u_n|^2- 2 \langle u_n, u \rangle +|u|^2$, so $$ \limsup_n |u_n-u|^2 \le \limsup_n |u_n|^2 -2 \liminf_n \langle u_n, u \rangle + |u|^2. $$
We have $\limsup_n |u_n| \le |u|$ implies $\limsup_n |u_n|^2 \le |u|^2$. We have $u_n \to u$ in $\sigma (H, H^*)$ implies $\liminf_n \langle u_n, u \rangle =|u|^2$. It follows that $$ \limsup_n |u_n-u|^2 \le 0. $$
This completes the proof.