Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals

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Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $a \in A$.

Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$ in $M$.


Then:

$\BB = \set {\map {B_{1/n}} a : n \in \N}$ is a local basis at $a$.


Proof

Let $\sequence {x_n}$ be the sequence in $\R$ defined as:

$x_n = \dfrac 1 n$

From Sequence of Reciprocals is Null Sequence, $\sequence {x_n}$ is a real null sequence.

From Null Sequence induces Local Basis in Metric Space:

$\BB = \set {\map {B_{1/n} } a : n \in \N}$ is a local basis at $a$.

$\blacksquare$