P-Sequence Space of Real Sequences 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 $d_p$ be the $p$-sequence metric on $\R$.

Then $\ell^p := \left({A, d_p}\right)$ is a metric space.

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

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

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

So axiom $M2$ holds for $d_p$.

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

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