Metric Subspace Induces Subspace Topology

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

Let $H \subseteq A$.

Let $\tau$ be the topology induced by the metric $d$.

Let $\tau_H$ be the subspace topology induced by $\tau$ on $H$.

Let $d_H$ be the subspace metric induced by $d$ on $H$.

Let $\tau_{d_H}$ be the subspace topology induced by $d_H$ on $H$.

Then:
 * $\tau_{d_H} = \tau_H$