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

Theorem

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

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

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

Let $a \in A$.

Then $a \in H^-$ if and only if there exists a sequence $\sequence {x_n}$ of points of $H$ which converges to the limit $a$.

From definition of closure, $H^- = H' \cup H^i$.

Suppose that $a \in H^-$.

If $a \in H^i$, then $a \in H$ and so $\sequence {a, a, \ldots}$ is a sequence in $H$ that converges to $a$.

If $a \in H'$, then by Definition of Limit Point (Metric Space) there exists a sequence in $H$ that converges to $a$.

$\Box$

For the converse, let $\sequence {x_n}$ be a sequence in $H$ that converges to a point $a \in A$.

So let $U$ be an open set that contains $a$.

By the definition of an open set, it follows that there exists an open ball centered at $a$ such that $\map {B_\epsilon} a \subseteq U$.

But then because $\sequence {x_n}$ converges to $a$ there exists $m \in \N_{\gt 0}$ such that:

$\ds \map d {x_m, a}$

$<$

$\ds \epsilon$

$\ds \leadsto \ \ $

$\ds x_m$

$\in$

$\ds \map {B_\epsilon} a$

Definition of Open Ball

$\ds \leadsto \ \ $

$\ds H \cap \map {B_\epsilon} a$

$\ne$

$\ds \O$

by hypothesis $\sequence {x_n}$ is sequence of points of $H$

$\ds \leadsto \ \ $

$\ds H \cap U$

$\ne$

$\ds \O$

$x_m \in \map {B_\epsilon} a$ and $\map {B_\epsilon} a \subseteq U$ leads to $x_m \in U$

Because from Set Intersection Preserves Subsets it is seen that $H \cap \map {B_\epsilon} a \subseteq H \cap U$.

It has been shown that every open set that contains $a$ also contains a point in $H$.

$\blacksquare$

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