Prove or disprove existence of a sequence converging weakly to $0$ in an infinite dim Hilbert space

Prove or disprove existence of a sequence converging weakly to $0$ in an
infinite dim Hilbert space

This is a problem on an old analysis qual, the prompt is:
"Prove or give a counter example: if $H$ is an infinite dimensional
Hilbert space and $0$ is the zero vector in $H$, then there exists a
sequence $\{x_n\}$ in $H$ so that $||x_n|| \ge 1$ and $\{x_n\}$ converges
weakly to the zero vector $0$ in $H$."
I know that the unit ball is not necessarily weakly compact in an infinite
dimensional space if it is not reflexive. But this is specifying the
existence of a single sequence, which doesn't say anything about every
sequence having a convergent subsequence etc.
Since it is a Hilbert space I know it is equivalent to $(x_n,y)
\rightarrow 0$ for all $y \in H$ for such a space. I was tempted to assume
a countable orthonormal use Parseval's Identity to show $||x_n||^2$ could
be made less than 1, but this would seem to require $(x_n,e_k)$ to
converge uniformly (ie independently of $k$ where the $e_k$'s are the
orthonormal set).
Anyway, I am stuck. Any suggestions? Thanks!