Hilbert Sequence Space is Metric Space/Proof 2

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

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.

Then $\ell^2 := \struct {A, d_2}$ where $d_2: A \times A: \to \R$ is the real-valued function defined as:
 * $\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$

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

Proof of $\text M 1$
So axiom $\text M 1$ holds for $d_2$.

Proof of $\text M 2$
Let $z = \sequence {z_i} \in A$.

So axiom $\text M 2$ holds for $d_2$.

Proof of $\text M 3$
So axiom $\text M 3$ holds for $d_2$.

Proof of $\text M 4$
So axiom $\text M 4$ holds for $d_2$.