Is it true that, if $\{u_n\}$ is a sequence in a Hilbert space $H$ that converges weakly to its limit $u \in H$ and the sequence satisfies $$\limsup_{n \rightarrow \infty} \|u_n \| \leq \|u\|$$ then $$\lim_{n \rightarrow \infty} \|u_n \| = \|u\|.$$
My attempt:
I wanted to show $$ \|u\| \le \liminf_{n \to \infty} \|u_n\|, $$ if this is true, then combined with $\liminf \|u\| \le \limsup \|u\|$ the desired result follows.
If the forward implication is true, then to my understanding this would in fact be an "if and only if" relation because the converse implication is trivial.