This is actually a problem from Schaum's Outline Series: General Topology.
Let $\{u_n\}$ be a sequence of real numbers such that the series $$\sum_{n=1}^\infty u_n^2<\infty.$$ That is, the series $u_1^2+u_2^2+\cdots$ converges. The class of all such sequences is denoted by $\mathbb R^\infty$.
The $l_2$-metric $d$, is defined on $\mathbb R^\infty$ as $$d(u,v)=\sqrt{\sum_{n=1}^\infty |u-v|^2}.$$
$\mathbb R^\infty$ with the $l_2$-metric is called the $l_2$-space (or Hilbert space) $H$.
Show that $H$ is second countable and separable.
I think I'll be fine if I'll be given the countable base (for second countability) and countable dense subset (for separability) and I can work on the rest.
Best Answer
A separable metric space is automatically second countable, so it's enough to show that $\ell^2$ is separable. I'd start by looking at the set of sequences of rational numbers which are eventually zero.