We are given a metric space $(M,d(x,y))$ and a sequence $\{x_j\}_{j=1}^\infty$ of elements of a compact set $L\subseteq M$. Also K is the set of $x\in M$ for which there is a subsequence, $\{x_{j_n}\}_{n=1}^\infty$ that converges to $x$. If $K$ has exactly one element, then we have to show that $\{x_j\}_{j=1}^\infty$ converges to that element.
Is this problem the same as proving that if all possible subsequences of a given sequence of elements of a compact set converge to an element, then the sequence also converges to that same element?
And if not, how do we prove this result? By contradiction?