Hilbert Sequence Space is Separable

Theorem
Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.

Then $\ell^2$ is a separable space.

Proof
Consider the set $H$ of all points of $\ell^2$ which have finitely many rational coordinates and all the rest zero.

$H$ is countable, since
 * Rational Numbers are Countably Infinite
 * Cartesian Product of Countable Sets is Countable
 * Countable Union of Countable Sets is Countable

It remains to show that $H$ is everywhere dense in $\ell^2$.

Indeed, the result then follows by definition of separable space.

To this end, let $\sequence {x_i} \in \ell^2$.

For each $n \in \N$, define $\sequence {q^{\paren n} _i} \in H$ by:
 * $q^{\paren n} _i :=

\begin {cases} \frac {\floor {2^{n+i} x} }{2^{n+i} } & : i < n \\ 0 & : i \ge n \end{cases}$ where $\floor \cdot$ denotes the floor function.

Then, we have: