1. The problem statement, all variables and given/known data (0,100) has a cover that consists of a finite number of closed interval subsets. I'm really lost with this one. I can clearly understand why the statement is false, but I'm not sure my proof is good. 2. Relevant equations 3. The attempt at a solution Clearly this is false, so I am trying to disprove it. Proof: Let S =(0,100) and let C be a cover of S. Since C contains finitely many closed interval subsets of S, C has a least element. Let X_i =[a,b][itex]\in[/itex]C be a subset of S, [itex]\forall[/itex]a,b[itex]\in[/itex]R^+ such that 0<a<b<100 Since there is no smallest positive real number, (which i have proved before), there is an infinite number of X_i's in C. But C has a finite number of elements. This is a contradiction. Q.E.D.