A question about compactness

Question

There is the question: deduce that $Y$ is a compact subset of $(X,d)$ iff the metric space $(Y,d)$ is compact. (Given that $Y$is a subset of $X$).

How to show it? (I cannot find anything to show though...)

Answer 1

Be sure you are using a definition of compactness that does not depend on the ambient space, such as the traditional one that any open cover admits a finite subcover.

Answer 2

I suppose that the context is this one: we say that $Y(\subset X)$ is compact if, for every family $(A_\lambda)_{\lambda\in\Lambda}$ such that $Y\subset\bigcup_{\lambda\in\Lambda}$, there is a finite subset $F$ of $\Lambda$ such that $Y\subset\bigcup_{\lambda\in F}$. If so, saying that $(Y,d)$ is a compact space and saying that $Y$ is a compct subset of $X$ are two different things. It's not hard to prove that they are equivalent.