# Closure of Subset of Closed Set of Metric Space is Subset/Proof 2

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## Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $F$ be a closed set of $M$.

Let $H \subseteq F$ be a subset of $F$.

Let $H^-$ denote the closure of $H$.

Then $H^- \subseteq F$.

## Proof

Let $x \in H^-$.

From Point in Closure of Subset of Metric Space iff Limit of Sequence

By assumption:

- $\sequence {a_n}$ is also a sequence of points of $F$

From Subset of Metric Space contains Limits of Sequences iff Closed:

- $x \in F$

Thus it has been shown:

- $H^- \subseteq F$

$\blacksquare$

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.6$: Open Sets and Closed Sets: Exercise $6$