5.26 (Brezis) Prove that $\sqrt{n}u_n\rightharpoonup 0$ weakly.

Question

5.26 (Brezis) Assume that $(e_n)$ is a orthonormal basis of H.

1. Check that $e_n\rightharpoonup 0$ weakly.

Let $(a_n)$ be a bounded sequence in $\mathbb{R}$ and set $u_n=\frac{1}{n}\displaystyle\sum_{i=1}^{n}a_i e_i$.

2. Prove that $|u_n|\rightarrow 0$.

3. Prove that $\sqrt{n}u_n\rightharpoonup 0$ weakly.

DOUBT: Items 1 and 2 are OK. However, I'm struggling with item 3.

MY ATTEMPT: I know that in this case, $H$ is separable and isomorphic to $\ell^2$ because there is an orthonormal subspace of $H$. And doing some sketches we can find $$\|\sqrt{n}u_n\|^2=\frac{1}{n} \sum_{i=1}^{n}|a_n|^2 \leq M^2$$ where the last inequality is due to item 2. However, I wouldn't prove that $\sqrt{n}u_n$ converges pointwise to $0$ (zero), and why this implies that weak convergence which was required.