Hi,

I am trying to understand the first proof "A setSof real numbers is compact if and only if every open coverCofScan be reduced to a finite subcovering. " on this page: Theorem 5.2.6: Heine-Borel Theorem

But I don't understand towards the end when they say "However,aN+1is an element ofS, so that this subcovering can not coverS." How do we knowaN+1is an element ofS?