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

Show that a subspace X of the Euclidean space Rⁿ 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.

Originally Posted by Cloud141

Show that a subspace X of the Euclidean space Rⁿ 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?