Hilbert Sequence Space is Lindelöf

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

Then $\ell^2$ is a Lindelöf space.

Proof
From Hilbert Sequence Space is Second-Countable, $\ell^2$ is a second-countable space.

The result follows from Second-Countable Space is Lindelöf.