Let ${e_n}$ be an orthonormal basis for $l^2$ and ${alpha_n} in ell^{infty}$.

Define $Ae_n=alpha_n e_n$ on $l^2$.

Find the condition for $A$ to be compact.

My work:

I want to show that in order for $A$ to be compact, we must have $alpha_n rightarrow 0$.

Suppose the contrary.

Then there exists some $epsilon >0 $ such that we can find some sequence $(alpha_{n_k})_k$ such that $(alpha_{n_k})_k> epsilon$.

I want to show that in this case, we can’t find a convergent subsequence for $(Ae_{n_k})=(alpha_{n_k}e_n)_k$.

And I got stuck here. Not sure how to show that there doesn’t exist a convergent subsequence.

I feel like I’m missing some facts in real analysis.

Any help is appreciated! Thank you.