Hilbert Sequence Space is Second-Countable

Theorem
Let $\ell^2$ be the Hilbert sequence space on $\R$.

Then $\ell^2$ is a second-countable space.

Proof
From Hilbert Sequence Space is Separable, $\ell^2$ is a separable space.

We also have that Hilbert Sequence Space is Metric Space.

The result follows from Separable Metric Space is Second-Countable.