1. The problem statement, all variables and given/known data Let K be a subset of R. Prove that if every sequence in K has an accumulation point, then K must be compact. 2. Relevant equations I tried to proof it below. Am I on the right track? 3. The attempt at a solution My intuition; Let x_n be sequence in K whose accumulation point is x, then there is a sub-sequence. x_n_k converges to x. Since sub-sequence x_n_k converges to x, x_n_k is a Cauchy sequence. We know that Cauchy sequences are bounded. Thus x_n is bounded. so K is bounded. Since all accumulation points are in K so K is closed. By Heine-Borel Thm, K is compact.