Given a real $x$ and an integer $N \gt 1$, prove that there exist integers $h$ and $k$ with $0 \lt k \le N$ such that $|kx-h|\lt 1/N$. Given a real $x$ and an integer $N \gt 1$, prove that there exist integers $h$ and $k$ with $0 \lt k \le N$ such that $|kx-h|\lt 1/N$. Hint. Consider the $N+1$ numbers $tx-[tx]$ for $t=0,1,2,\dots, N$ and show that some pair differs by at most $1/N$.

I'm trying to prove this hint. Let $a_t=tx-[tx]$ for $t=0,1,2,\dots, N$, and assume that for every different pair $|a_i-a_j| \ge 1/N$. 

I can't lead to a contradiction from this. I would greatly appreciate any help.
PLEASE delete, I misread the (conditions of the) OP.
PLEASE delete, I misread the (conditions of the) OP.