AoPS - Problem

Source: IMO LongList 1967, The Democratic Republic Of Germany 4

Tags: number theory, approximation, irrational number

orl

16.12.2004 23:11

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

Okay...this solution may not be correct, please tell me where I'm wrong. We are given that r1/r2 is irrational, which implies that r1 and r2 are both irrational. Now, put z=k1r1 +k2r2. Choosing r1, and r2 as a base, we can construct a coordinate system so that any number can be written in the form of z. So, any number p can be written as the absolute value of the difference of two numbers. That number can be as small as we please. Hence the proof. Is this correct?! Sorry for the bad presentation....how does this latex thing work?