= Solution
The hypothesis says exactly that $X_n\rightharpoonup0$ in the <Hilbert space> $L^2$. We give the constructive <weak Banach–Saks theorem in a Hilbert space> argument. Choose $n_1<n_2<\cdots$ inductively so that, for $Y_k=X_{n_k}$,
$$
|\mathbb E(Y_jY_k)|\leq2^{-k}
\qquad(1\leq j<k).
$$
This is possible because each earlier $Y_j$ is an admissible test variable in the weak-convergence assumption. Since $\lVert Y_k\rVert_2\leq1$,
$$
\begin{aligned}
\left\lVert\frac1N\sum_{k=1}^NY_k\right\rVert_2^2
&=\frac1{N^2}\left(
\sum_{k=1}^N\lVert Y_k\rVert_2^2
+2\sum_{1\leq j<k\leq N}\mathbb E(Y_jY_k)
\right)\\
&\leq\frac1N+
\frac2{N^2}\sum_{k=2}^N(k-1)2^{-k}
\longrightarrow0.
\end{aligned}
$$
Therefore
$$
\boxed{\frac1N\sum_{k=1}^NY_k\longrightarrow0
\quad\text{in }L^2.}
$$
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