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

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Let $a \in A$.

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

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