1. The problem statement, all variables and given/known data Theorem: In a metric space X, if (xn) is a Cauchy sequence with a subsequence (xn_k) such that xn_k -> a, then xn->a. 2. Relevant equations N/A 3. The attempt at a solution 1) According to this theorem, if we can show that ONE subsequence of xn converges to a, is that enough? Or do we need to show that EVERY subseuqence of xn converges to a in order to claim that xn->a? 2) How can we prove the theorem? By definition, xn is Cauchy iff for all ε>0, there exists N s.t. if n,m≥N, then d(an,am)<ε. By definition, xn->a iff for all ε>0, there exists M s.t. if n≥M, then d(xn,a)<ε. I know the definitions, but I don't see how to link these all together to prove the theorem... Any help is greatly appreciated!