The problem statement, all variables and given/known data Show that every compact set must be closed. I am looking for a simple proof. This is supposed to be Intro Analysis proof. Relevant equations Any compact set must be bounded. 3. The attempt at a solution Suppose A is not closed, so let a be an accumulation point of A such that a is in A etc. (I think I can finish this proof, but I do not like it.) Does anyone know of a simpler proof.