Hello, I have a question about Heine Borel Theorem. First, I am not sure why we have to show "gamma=Beta" gamma is the supremum of F(which is equivalent to H_squiggly_bar in the text ), and it has to be greater than beta. Otherwise, S contains H_squiggly_bar Second, for the case 1, why S_gamma+eps does not have a finite subcovering? which definition the author is reffering to? I understand sup(F) = gamma, so S_gamma-eps must have a finite subcovering because by definition H_squiggly_bar is a set of finite subcovering. But isn't there a possibility that S_gamma+eps also has a finite subcovering? That consists of H_squiggly bar + some finite set that belongs to H but not contained within H_squiglly bar?