Compactness of $A \subset l^2(\mathbb{N})$

Question

I'm wondering about how to approach something like this:

Let $c \in \ell^2(\mathbb{N})$, $A = \{x \in \ell^2(\mathbb{N}) : |x(n)| \le |c(n)|, \forall n \in \mathbb{N}\}$.

Show that $A$ is compact in $\ell^2(\mathbb{N})$.

Should I invoke Tichonov theorem?

Answer 1

Hint: Asserting that $A$ is compact is equivalent ot asserting that it is both complete and totally bounded. But, since $\ell^2(\mathbb{N})$ is complete and $A$ is closed, it is clear that it is complete. So, all that remains to be proved is that it is totally bounded.