Subspace of Metrizable Space is Metrizable Space

Theorem
Let $\struct{X, \tau}$ be a metrizable topological space.

Let $\struct{Y, \tau_Y}$ be a subspace of $\struct{X, \tau}$.

Then:
 * $\struct{Y, \tau_Y}$ is a metrizable topological space

Proof
By definition of metrizable topological space:
 * there exists a metric $d$ on $X$ such that the topology induced by $d$ is $\tau$

From Metric Subspace Induces Subspace Topology:
 * $\tau_Y$ is the topology induced by the subspace metric $d_Y$ on $Y$

From Subspace of Metric Space is Metric Space:
 * $\struct{Y, d_Y}$ is a metric space

By definition, $\struct{Y, \tau_Y}$ is a metrizable topological space.