Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M4

Theorem
Let $\alpha$ be an infinite cardinal number.

Let $I$ be an indexed set of cardinality $\alpha$.

Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
 * $(1)\quad \set{i \in I: x_i \ne 0}$ is countable
 * $(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.

Let $d_2: A \times A: \to \R$ be the real-valued function defined as:
 * $\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$

Then $d_2$ satisfies.

Proof
Let $H = \struct{\ell^2, d_{\ell^2}}$ denote the Hilbert sequence space, where:
 * $\ell^2$ denotes the real $2$-sequence space, that is, the set of all real sequences $\sequence {x_n}$ such that the series $\ds \sum_{n \mathop = 0}^\infty x_n^2$ is convergent
 * $d_{\ell^2}$ denotes the real $2$-sequence metric, that is, the real-valued function $d_{\ell^2}: \ell^2 \times \ell^2: \to \R$ defined as:
 * $\ds \forall x = \sequence {x_n}, y = \sequence {y_n} \in \ell^2: \map {d_{\ell^2}} {x, y} := \paren {\sum_{n \mathop = 0}^\infty \paren {x_n - y_n}^2}^{\frac 1 2}$

Lemma 3
Let $x_1, x_2 \in A : x_1 \ne x_2$.

From Lemma 3:
 * $\exists y_1, y_2 \in \ell_R^2 :$
 * $y_1 \ne y_2$
 * $\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$

We have:

The result follows.