Okay I have been asked to prove that Every compact subspace of a metric space is closed and bounded.
The proof that it is closed is very easy as every metric space is a hausdorff space ,and using definitions we can show that it is closed (Which I have done)
For bounded ,what I am thinking is that ,Since Y(lets us call the compact subspace as Y) is compact , there will be an open cover for the following subspace consisting of open sets from that metric space . In a metric space the open sets are open balls of given radius . Since the balls are of a given radius,the union of all these balls…