Find a sequence of elements $(x_n)$ in a Hilbert space converging weakly to $x$ with $\lim \inf_{n\rightarrow \infty} ||x_n|| > ||x||$

Question

This is a practice exam question and I am hoping to check whether my solution is correct.

Take the sequence of standard basis vectors $(e_n)$ in $l^2 (\mathbb{Z})$. Then, $(e_n)$ converges weakly to $0$, since $$\lim_{n\rightarrow\infty} \langle e_n, x\rangle = \langle \lim_{n\rightarrow \infty} e_n, x\rangle = \langle 0, x\rangle$$ for $x \in l^2(\mathbb{Z})$. Observe that $||e_n|| = 1$ for all $n$. Then, $\lim\inf ||e_n|| = 1 > ||0|| = 0$.