>>6962990

That fucking series proof smh

Define .999... as an upper bound of every subset [0, a] of the real line where 0<a<1. Denote by F the family of complements of every such set with respect to [0,1]. Then because [0,1] is clearly compact, the nested set property provides that the intersection of F contains precisely one element, .999... by hypothesis. But 1 is also an element of every such set in F. Hence, .999...=1.