Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.3 alpha 4/15/85; site ucbvax.ARPA
Path: utzoo!watmath!clyde!burl!ulysses!ucbvax!phr
From: phr@ucbvax.ARPA (Paul Rubin)
Newsgroups: net.math
Subject: Another head scratcher, and solution to the last one.
Message-ID: <7560@ucbvax.ARPA>
Date: Wed, 29-May-85 01:14:45 EDT
Article-I.D.: ucbvax.7560
Posted: Wed May 29 01:14:45 1985
Date-Received: Thu, 30-May-85 04:06:01 EDT
Organization: University of California at Berkeley
Lines: 30

Andrew Nash, Peter Royappa, and a few other people answered the last
head scratcher, which was to demonstrate a bijection between (0,1) and
[0,1].  The solutions were all pretty hard for me to follow, but the
idea was the same in each:  [0,1] is (0,1) with two extra points, 0 and 1.
So you can map everything to itself, except you have to first get
rid of these points, by mapping them to (say) 1/2 and 1/3 respectively.
Then you must get rid of 1/2 and 1/3, by sending them to (say)
1/4 and 1/5, and so on down the line.  One such mapping f from
[0,1] to (0,1) then is

	f(x) = x, unless x = 0 or x = 1/n for some integer n;
	f(x) = 1/(n+2) otherwise.

======================================================================
Now the new one:

Give an example of an infinite sequence a_0, a_1, ... so that the series

infinity
 ____
 \     a_n   does not converge, but
 /___
 n = 0


infinity
 ____
 \     a_n / (1 + n a_n)      converges.
 /___
 n = 0