# real analysis – condition for compactness of an operator

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.