From: "William L. Bahn" Subject: Re: All positive integers are equal Date: Wed, 1 Dec 1999 23:57:40 -0700 Newsgroups: sci.math For k=1, you have four possibilities, {r,s} = {1,1}, {1,2}, {2,1}, {2,2} In the first and last case, r=s. But in ALL of the cases where r#s, you end up with at least one 0 when you take r-1 and s-1. This is outside the constraint that you are working with (strictly) positive integers. Jim Ferry <"jferry"@[delete_this]uiuc.edu> wrote in message <824bmk$ph7$3@vixen.cso.uiuc.edu>... >I like this proof. It's probably quite old, and I'll accept being >flamed for posting it, but I'd never seen it before, so here it goes: > >Proof that all positive integers are equal: > >It suffices to show that for any two positive integers r and s, >r = s. We proceed by induction. Let P(n) be the statement that >all positive integers <= n are equal. P(1) is clearly true. If >P(k) is true, then consider any two integers r and s, each <= k+1. >Then r-1 and s-1 are each <= k, so r-1 = s-1. Hence r = s, which >proves P(k+1). > >| Jim Ferry | Center for Simulation | >+------------------------------------+ of Advanced Rockets | >| http://www.uiuc.edu/ph/www/jferry/ +------------------------+ >| jferry@[delete_this]uiuc.edu | University of Illinois |