Prove the following statement: If
and
are real numbers whose quotient is irrational, then any real number
can be approximated arbitrarily well by the numbers of the form
integers, i.e. for every number
and every positive real number
two integers
and
can be found so that
holds.
%V0
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.
Školjka