P-Sequence Space of Real Sequences is Metric Space

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 $d_p$ be the $p$-sequence metric on $\R$.

Then $\ell^p := \struct {A, d_p}$ is a metric space.

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

Proof of
So holds for $d_p$.

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

So holds for $d_p$.

Proof of
So holds for $d_p$.

Proof of
So holds for $d_p$.