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

Theorem
Let $M = \left({A, d}\right)$ 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:
 * $H' \cap H^i = \varnothing$

Proof
Let $a \in H_i$.

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

But by metric space axiom $M1$:
 * $d \left({a, a}\right) = 0$

and so:
 * $\left\{{x \in H: 0 < d \left({x, a}\right) < \epsilon}\right\} = \varnothing$

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

That is:
 * $a \notin H'$

or:
 * $a \in \complement_A \left({H'}\right)$

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