Density of Q diluted?

kindlychung

[这个贴子最后由kindlychung在 2010/10/04 01:26pm 第 1 次编辑]

elimqiu

Given reals a, b with a < b,
let p be an odd number such that (b-a)p > 2
let k be an integer such that (k-1)/p ≤ a ＜k/p
then k/p - a ≤ 1/p < (b-a)/2 and so  a ＜k/p < (a+b)/2
and so a < k/p < (k+1)/p < b
let m'; be the even one of k and k+1, we have a < m';/p < b
let d = gcd(m';,p) ( greatest common dividor of m'; and p)
m = m';/d, n = p/d, then gcd(m,n) = 1 and since d is odd,
we see m is even and a < m/n < b
Therefore Q1 is dense in R

kindlychung

Neat. Thank you.

wangyangkee

外文帖子，别往下掉，，，