Squeeze Theorem/Sequences/Metric Spaces

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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 $p$.


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