Re: A set dense in R
From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 12/28/04
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Date: Tue, 28 Dec 2004 03:00:08 GMT
In sci.math, Amanda
<sca18@hotmail.com>
wrote
on 27 Dec 2004 12:16:09 -0800
<6f75d9cf.0412271216.657c40d7@posting.google.com>:
> Hello
>
> I'd like some hints to prove that, for every real p<>0, the set A =
> {sqrt(n) + m*p | n>=0 is integer, m is integer} is dense in R. I
> tried to use the pigeon hole principle, but got stuck.
> Thank you.
Simplest method I can think of is to prove it dense around k,
for any 0 <= k < p, by looking at the expression
sqrt(n) - floor((sqrt(n) - k)/p)*p
and noting what its lower bound is as n -> +oo.
-- #191, ewill3@earthlink.net It's still legal to go .sigless.
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