Q:

Find an approximation of the unit ball as a weak-limit of a sequence in the unit sphere

Let $H$ be an infinite dimensional Hilbert space.
It is well known that the weak-closure of the unit ball is the unit sphere. But I want to prove it as basicaly as possible, using the weakly-sequential deffinition of closure, by actually finding a convering sequence of members of B.
It is clear to me that $\overline S \subset B$ since $B$ is strong-closed and convex.
For the hard and interesting part, I figured out that I could pick an orthonormal basis for $H$,$\{e_a\}_{a\in A}$, and given a $x\in B$ write it as $x=\sum_{n=1}^\infty \langle x,e_n\rangle e_n$ for $(e_n)_{n=1}^{\infty} \subset \{e_a\}_{a\in A}$.
Couldn't go any further.
Please guide me a bit, and let me know if you think that what I'm trying to do is possible. A nice hint would be welcomed :)
Thanks!

A:

Fix $x\in H$ with $\|x\|\leq1$. Now define
$$
x_n=\sum_{k=1}^n\langle x,e_n\rangle\,e_n\ +(1-\sum_{k=1}^n|\langle x,e_n\rangle|^2)^{1/2}e_{n+1}.
$$
Then $\|x_n\|=1$ for all $n$.
For any $y\in H$,
$$
\langle x-x_n,y\rangle=[\langle x,e_n\rangle-(1-\sum_{k=1}^n|\langle x,e_n\rangle|^2)^{1/2}]\,\bar y_n+\sum_{k=n+1}^\infty \langle x,e_n\rangle\bar y_n\leq2\left(\sum_{k=n}^\infty|y_n|^2\right)^{1/2}\to0
$$

