Suppose $H$ is a Hilbert space with an orthonormal basis $(e_n)$. Let $(x_n)$ be a bounded sequence in $H$.
- Prove that $(x_n)$ contains a subsequence $(x_{n_k})$ and $H$ contains an element $x$ so that $\langle x_{n_k},e_n\rangle\to\langle x,e_n\rangle$ for each $n$.
- Prove that $\langle x_{n_k},x'\rangle\to\langle x,x'\rangle$ for any $x'\in H$.
I am not sure how to proceed with this question. I am familiar wiht the Riesz representation theorem, which states that for every $f\in H^*$ there exists $y\in H$ so that $f(x)=\langle x,y\rangle$. But I don't know how to apply that here.
I tried defining a linear functional $f(h)=\langle h,e_n\rangle$ and seeing what I can do from there, but I am truly stuck. I have seen some proofs using the Banach-Alaoglu theorem, but I am wondering if there is a proof without using this result. Any help in this question will be highly appreciated - thanks in advance.