Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals
Jump to navigation
Jump to search
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 $a$.
$\blacksquare$