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
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Exercise $6$