Set of Isolated Points of Metric Space is Disjoint from Limit Points

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

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

Let $H'$ be the set of limit points of $H$.

Let $H^i$ be the set of isolated points of $H$.

Then:
 * $H' \cap H^i = \O$

Proof
Let $a \in H_i$.

Then by definition of isolated point:
 * $\exists \epsilon \in \R_{>0}: \set {x \in H: \map d {x, a} < \epsilon} = \set a$

But by :
 * $\map d {a, a} = 0$

and so:
 * $\set {x \in H: 0 < \map d {x, a} < \epsilon} = \O$

So by definition $a$ is not a limit point of $H$.

That is:
 * $a \notin H'$

or:
 * $a \in \relcomp A {H'}$

It follows from Intersection with Complement is Empty iff Subset that:
 * $H' \cap H^i = \O$