From: "William L. Bahn" <wbahn@uswest.net>
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 |