Squeeze Theorem/Sequences/Metric Spaces
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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 $p$.
$\blacksquare$