Isometry between Metric Spaces is Continuous

Theorem
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $\phi: M_1 \to M_2$ be an isometry.

Then $\phi: M_1 \to M_2$ is a continuous mapping.

Proof
Let $a \in A_1$.

Let $\epsilon \in \R_{>0}$.

Let $\delta = \epsilon$.

Then:

So by definition $\phi$ is continuous at $a$.

As $a \in H$ is arbitrary, it follows that $d_H$ is continuous on $H$.