# 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 $p$.

## Proof

 $\ds \forall n \in \N: \,$ $\ds \map d {p, x_n}$ $\le$ $\ds r_n$ by hypothesis $\ds \forall n \in \N: \,$ $\ds r_n$ $\le$ $\ds \size {r_n}$ Negative of Absolute Value $\ds \forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: \,$ $\, \ds n > N \implies \,$ $\ds \size {r_n}$ $<$ $\ds \epsilon$ Definition of Null Sequence $\ds \leadsto \ \$ $\ds \forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: \,$ $\, \ds n > N \implies \,$ $\ds \map d {p, x_n}$ $<$ $\ds \epsilon$ Extended Transitivity

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

$\blacksquare$