Here is an attempt. It seems you are not trying to use Heine-Borel theorem. Which makes is harder. Let be defined as in the hint. This set is non-empty because . And it has an upper bound, ii.e. . Thus, by completeness there is a least upper bound . Note, , we will show is impossible. Let be an open covering of . By construction of it means there exists finitely many open sets which cover . So it means without lose of generality. But since it is an open set it means there is such that . Also since it means there is such that . Let . Then by construction and can be covered too. So . Which contradicts that is the least upper bound. Thus, it means . And so is covering compact.