User:Leigh.Samphier/Topology/Nagata-Smirnov Metrization Theorem/Lemma 2

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $\BB = \ds \bigcup_{n \mathop \in \N} \BB_n$ be a $\sigma$-locally finite basis of $T$ where $\BB_n$ is locally finite set of subsets for each $n \in \N$.

Let $I = \set{\tuple{B, n} : B \in \BB, B \in \BB_n}$.

For each $\tuple{B, n} \in I$, let $f_{\tuple{B, n}}:S \to \closedint 0 1$:
 * $B = \set{x \in S : \map {f_{\tuple{B, n}}} x \ne 0}$

Let $g_n : S \to \closedint 0 1$ be the mapping defined by:
 * $\map {g_n} x$ is the limit of the generalized sum $\ds \sum_{B \in \BB_n} \map {f_{\tuple{B, n}}^2} x$

Then:
 * for all $x \in S$:
 * the generalized sum $\ds \sum_{\tuple{B, n} \mathop \in I} \sqbrk{\dfrac 1 {\paren{\sqrt 2}^n} \dfrac {\map {f_{\tuple{B, n}}} x} {\sqrt {1 + \map {g_n} x}}}^2$ converges

Proof
Let $\FF$ denote the set of finite subsets of $I$.

Let $F \in \FF$.

Hence:
 * $\set{n \in \N : \exists B \in \BB_n : \tuple{B, n} \in F}$ is finite.

Let $\set{n_1, n_2, \ldots, n_m} = \set{n \in \N : \exists B \in \BB_n : \tuple{B, n} \in F}$.

We have:

Since $F$ was arbitrary, it follws that:
 * $\forall F \in \FF : \ds \sum_{\tuple{B, n} \mathop \in F} \sqbrk{\dfrac 1 {\paren{\sqrt 2}^n} \dfrac {\map {f_{\tuple{b, n} } } x} {\sqrt {1 + \map {g_n} x} } }^2 \le 2$

From User:Leigh.Samphier/Topology/Bounded Generalized Sum is Absolutely Convergent:
 * the generalized sum $\ds \sum_{\tuple{B, n} \mathop \in I} \sqbrk{\dfrac 1 {\paren{\sqrt 2}^n} \dfrac {\map {f_{\tuple{B, n}}} x} {\sqrt {1 + \map {g_n} x}}}^2$ converges