Since x is irrational there exists an integer N for which
N<x<N+1. Divide the interval [N,N+1] into n2 parts as
xj=N+j/n2 for j=0,1,2,...,n2, (xn2=N+1)
All of the xj are rational. xj+1-xj=1/n2. No xj is x.
There is a j* such that xj*<x<xj*+1 That is a rational
which is closer than 1/n2 to x. Remember that xj=N+j/n2 = (n2N+j)/n2