Null Sequence induces Neighborhood Basis of Closed Sets in Metric Space
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\sequence {x_n}$ be a real null sequence such that:
- $\forall n \in N: x_n > 0$
Let $\map {B^-_\epsilon} a$ denote the closed $\epsilon$-ball of $a$ in $M$.
Then:
- $\NN_{\sequence {x_n} } = \set{\map {B^-_{x_n} } a : n \in \N}$ is a neighborhood basis of $a$ consisting of closed sets.
Proof
From Closed Ball in Metric Space is Closed Neighborhood:
- every element of $\NN_{\sequence {x_n} }$ is an closed neighborhood of $a$.
By definition of closed neighborhood of $a$:
- every element of $\NN_{\sequence {x_n} }$ is an neighborhood of $a$ and a closed set.
Let $U$ be an open neighborhood of $a$.
By definition of an open set, there exists a strictly positive real number $\epsilon$:
- $\map {B_\epsilon} a \subseteq U$
From Open Ball contains Strictly Smaller Closed Ball:
- $\map {B_{\epsilon/2}^-} a \subseteq \map {B_\epsilon} a$
By definition of a real null sequence:
- $\exists N \in \N : \forall n > N : \size {x_n} < \dfrac \epsilon 2$
Let $m = N + 1$.
Then $\size {x_m} < \dfrac \epsilon 2$.
Since $x_m > 0$ then $x_m = \size {x_m} < \dfrac \epsilon 2 $, and so:
- $\map {B_{x_m}^- } a \subseteq \map {B_{\epsilon/2}^-} a \subseteq \map {B_\epsilon} a \subseteq U$
The result follows from Subset Relation is Transitive.
$\blacksquare$