Marge -- thanks for reading my blog and your comment, but not sure I'm following. I understand "friendless" to mean having zero friends. Are you suggesting someone can have a negative number of friends? The proof is achieved via the Pigeonhole Principle, that if you stuff p pigeons into p-1 pigeonholes, at least one of the pigeonholes will have more than one pigeon. I just pray that the pigeonholes are roomy enough so that no pigeons are harmed during the aforesaid stuffing.