MathDB

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.

Problem Statement