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If Wikimedia had invented its "multi-login" setup before I had created this account, I would not be in this predicament, but alas, it was not to be. I am known under two (2) usernames. At the risk of being accused as a sock-puppeteer, I make this known here. The reason is that this username is taken on a few sister sites, so I use an alternate name there, but once the multi-login came about, My alternate obtained similarly-named accounts here and elsewhere. As a result, I now contribute using both users, as whim strikes me, but mostly my new one, since it does not have any name clashes. So look not here, but here for my recent contributions.
[edit] Macro Polo
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[edit] Destroyer 45
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[edit] A Proof that π really is irrational
Theorem: 
is irrational
Proof: Suppose 
, where 
and 
are integers. Consider the functions 
defined on ![[0, \pi]\!](//upload.wikimedia.org/wikipedia/en/math/5/8/a/58ae49f6a7bbbf8fda3b9c8a807008d1.png)
by
Clearly 
for all 
. Let 
denote the 
-th derivative of . Note that
; otherwise some integer
By repeatedly applying integration by parts, the definite integrals of the functions 
can be seen to have integer values. But are strictly positive, except for the two points 0 and , and these functions are bounded above by 
for all sufficiently large n. Thus for a large value of n, the definite integral of is some value strictly between 0 and 1, a contradiction.
[edit] Userboxes
is irrational
, where 
and 
are integers. Consider the functions 
defined on ![[0, \pi]\!](http://upload.wikimedia.org/wikipedia/en/math/5/8/a/58ae49f6a7bbbf8fda3b9c8a807008d1.png)
by
for all 
. Let 
denote the 
-th derivative of 
; otherwise some integer
can be seen to have integer values. But 
for all sufficiently large n. Thus for a large value of n, the definite integral of 
