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Closure of a Set in a Metric Space is all its Sequential Limits

Last updated Nov 1, 2022

# Statement

Suppose $(X, d)$ is a Metric Space

. Suppose

S \subset X

Then

\text{cl} S = {x : \exists (x_n) \subset S \text{ s.t. } x_{n} \to x}

# Proof

Metric Spaces are First Countable and Hausdorff and Closure of a Set in a First Countable Space is all its Sequential Limits

.

\blacksquare

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