Problem

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

Tags: number theory, approximation, irrational number



Prove the following statement: If r1 and r2 are real numbers whose quotient is irrational, then any real number x can be approximated arbitrarily well by the numbers of the form  zk1,k2=k1r1+k2r2 integers, i.e. for every number x and every positive real number p two integers k1 and k2 can be found so that |x(k1r1+k2r2)|<p holds.