Solution
= Solution
If $x=0$, the claim is immediate. Otherwise weak convergence gives
$$
\lVert x\rVert^2
=\langle x,x\rangle
=\lim_{n\to\infty}\langle x_n,x\rangle.
$$
By Cauchy-Schwarz,
$$
\langle x_n,x\rangle
\leq\lVert x_n\rVert\lVert x\rVert.
$$
Taking the lower limit and dividing by $\lVert x\rVert$ proves the <weak lower semicontinuity of the Hilbert norm>:
$$
\boxed{\lVert x\rVert\leq\liminf_n\lVert x_n\rVert}.
$$