Metric subspace

Definition

Given a metric space [ilmath](X,d)[/ilmath] and any [ilmath]A\subset X[/ilmath], we can define a metric as follows:

$d_A:A\times A\rightarrow\mathbb{R}$ where $d_A(x,y)\mapsto d(x,y)$ (so a restriction of the function essentially)

Then [ilmath](A,d_A)[/ilmath] is a metric subspace of [ilmath](X,d)[/ilmath] and [ilmath]d_H[/ilmath] is the induced metric.

TODO: proof it is a metric space