= Solution
For each $n$, define the bounded linear functional
$$
T_n:H\longrightarrow\mathbb C,
\qquad T_n(y)=\langle x_n,y\rangle.
$$
The hypothesis is precisely that $x_n$ converges <weak convergence in a Hilbert space>[weakly] to $x$. In particular, for each fixed $y\in H$, the scalar sequence $T_n(y)$ converges and is therefore bounded. Thus the family $\{T_n\}$ is pointwise bounded.
The <Uniform boundedness principle> gives
$$
\sup_n\lVert T_n\rVert<\infty.
$$
By the <Riesz representation theorem>, $\lVert T_n\rVert=\lVert x_n\rVert_H$. Hence
$$
\sup_n\lVert x_n\rVert_H<\infty,
$$
so the sequence is bounded.
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