About a hard question on PIgeon Hole Principle!?

Question

Let r be any irrrational real number. Prove that there exists a positive integer n so that the distance of nr from the closest integer is less than 10 (-10) = ( 10 the power minus 10)

Answer 1

Take as your buckets the half-open intervals [k*10^-10, (k+1)*10^-10) where k = 0, ... , 10^10-1.

If n is an integer, then the fractional part of n*r is in one of these intervals. Also, these values are all distinct, because if the fractional parts of n*r and m*r are the same, then (n-m)*r is an integer, and r is rational.

So, for n=1,...,10^10 + 1, there are 10^10+1 different fractional parts for n*r, and 10^10 buckets, so two of the fractional parts must be in the same bucket:

For some m!=n, the fractional parts of m*r, and n*r are in
[k*10^-10,(k+1)*10^-10) for some k.

But then (m-n)*r is within 10^-10 of an integer.