Re: In need for a proof from the book.

In article <1182282308.708688.4010@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
ayalgelles@xxxxxxxxx wrote:

Is there a proof from the book that there is a square of a rational
between any two rationals? (Preferably with no reference to real
numbers)

Suppose 0 <= r < s <=1, where r and s are rational. There exists a
natural number n such that 2/n < s - r. Now hop from 0 to 1 along the
points 0, 1/n^2, 4/n^2, ..., (n-1)^2/n^2, 1. The maximum hop is the last
one = 1 - (n-1)^2/n^2 = (2n-1)/n^2 < 2/n. So somewhere along the way you
must land in (r,s). If 0 <= r < s and s > 1, find a natural number m
such that s/m^2 < 1 and apply the above to r/m^2 and s/m^2.
.