Squeeze Theorem/Sequences/Metric Spaces

Theorem

Let $M = \struct {S, d}$ be a metric space or pseudometric space.

Let $p \in S$.

Let $\sequence {r_n}$ be a null sequence in $\R$.

Let $\sequence {x_n}$ be a sequence in $S$ such that:

$\forall n \in \N: \map d {p, x_n} \le r_n$.

Then $\sequence {x_n}$ converges to $p$.

$\ds \forall n \in \N: \, $

$\ds \map d {p, x_n}$

$\le$

$\ds r_n$

$\ds \forall n \in \N: \, $

$\ds r_n$

$\le$

$\ds \size {r_n}$

$\ds \forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: \, $

$\, \ds n > N \implies \, $

$\ds \size {r_n}$

$<$

$\ds \epsilon$

Definition of Null Sequence

$\ds \leadsto \ \ $

$\ds \forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: \, $

$\, \ds n > N \implies \, $

$\ds \map d {p, x_n}$

$<$

$\ds \epsilon$

Thus $\sequence {x_n}$ converges to $p$.

$\blacksquare$