Prove that a sequence xn converges to x if and only if every subsequence of xn has a subsequence that converges to x.
I'm pretty stuck on this one. I know the "=>" is very easy, because every subsequence of a convergent sequence is convergent.
But what about the "<=" way? I have spent 4 hours trying this now and nothing I do seems to make it work. Anyone have any tips?
I am trying. XNn can have no convergent subsequence. Suppose it does. Then there is XNnk a subsequence and For any e<0, there is A such that |x-XNnk|<e if Nnk>A. But any subsequence will eventually contain Nn>A such that |x-xNn|>e, as you said. Is this correct? I am not sure how to use math symbols on here, but I hope you get the idea of what I'm saying.