Showing subspace is compact IFF any sequence of elements has a converging subsequece

Show that a subspace X of the Euclidean space R^{n} is compact if and only if any

sequence of elements of X has a converging subsequence. Remark: this statement

holds in the considerably greater generality of any metric space but the proof of this

more general result is quite involved.

Re: Showing subspace is compact IFF any sequence of elements has a converging subsequ

Quote:

Originally Posted by

**Cloud141** Show that a subspace X of the Euclidean space R^{n} is compact if and only if any

sequence of elements of X has a converging subsequence. Remark: this statement

holds in the considerably greater generality of any metric space but the proof of this

more general result is quite involved.

How do you define a combact subspace?