Hilbert Sequence Space is Metric Space

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

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

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

Proof
By definition of the Hilbert sequence space on $\R$:

Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Then $\ell^2 := \left({A, d_2}\right)$ where $d_2: A \times A: \to \R$ is the real-valued function defined as:
 * $\displaystyle \forall x = \left\langle{x_i}\right\rangle, y = \left\langle{y_i}\right\rangle \in A: d_2 \left({x, y}\right) := \left({\sum_{k \mathop \ge 0} \left({x_k - y_k}\right)^2}\right)^{\frac 1 2}$

From Convergence of Square of Linear Combination of Sequences whose Squares Converge we have that $\displaystyle \sum_{k \mathop \ge 0} \left({x_k - y_k}\right)^2$ does actually converge.

Proof of $M1$
So axiom $M1$ holds for $d_2$.

Proof of $M2$
Let $z = \left\langle{z_i}\right\rangle \in A$.

So axiom $M2$ holds for $d$.

Proof of $M3$
So axiom $M3$ holds for $d_2$.

Proof of $M4$
So axiom $M4$ holds for $d_2$.