Quote by Office_Shredder
If any subsequence of x_{n} converges to a, then x_{n} converges to a too. Rather, what you can conclude is a single subsequence of x_{n} converges to a (which is what sequential compactness is about)

This is not true consider a_n=(1)^n=1,1,1,1,1,1,...
Then a_{2n}=1 is a subsequence that converges to 1, but a_n does not converge at all nonetheless to 1.