# 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 $a$.

## Proof

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 Limit Point is Limit of Convergent Sequence there exist 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:

\(\displaystyle \map d {x_m, a}\) | \(=\) | \(\displaystyle <\) | \epsilon | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle H \cap \map {B_\epsilon} a\) | \(\ne\) | \(\displaystyle \O\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle H \cap U\) | \(\ne\) | \(\displaystyle \O\) |

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 $H$.

So by Condition for Point being in Closure, it is seen that $a \in H^-$.

$\blacksquare$

## Sources

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