Metric subspace

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

[math]d_A:A\times A\rightarrow\mathbb{R}[/math] where [math]d_A(x,y)\mapsto d(x,y)[/math] (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