Inclusion Mapping on Metric Space is Continuous

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

Let $\struct {H, d_H}$ be a metric subspace of $M$.

Then the inclusion mapping $i_H: H \to A$ is continuous.

Proof
Let $a \in H$.

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

Let $\delta = \epsilon$.

Then:

So by definition $d_H$ is continuous at $a$.

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