Metrizability is hereditary

This article gives the statement, and possibly proof, of a topological space property (i.e., metrizable space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)
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Statement

If $X$ is a metrizable space, and $Y$ is a subset of $X$ given the subspace topology, then $Y$ is also a metrizable space.

Facts used

1. Topology from subspace metric equals subspace topology

Proof

Suppose $d$ is a metric on $X$ inducing the given topology on $X$. By fact (1), the subspace metric on $Y$ induces the subspace topology on $Y$. Thus, the subspace topology is induced by a metric, hence $Y$ is metrizable.