1. The problem statement, all variables and given/known data Suppose X ⊂ R^n is a compact set, and U_1, U_2, U3, ... ⊂ R^n are open sets whose union contains X. Prove that for some n ∈ N (the natural numbers) we have X ⊂ U_1 ∪ ... ∪ U_n. 2. Relevant equations A set is called compact if it is both closed and bounded. 3. The attempt at a solution This problem seems trivial to me. If, as stated in the problem, U_1, U_2, U3, ... ⊂ R^n are open sets whose union contains X, does that mean that for some n we have X ⊂ U_1 ∪ ... ∪ U_n? I don't understand how there is anything to prove here. Any help would be appreciated.