CMO 2010 Problem 1

Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that $$|T| \leq 1+ \frac{a_n-a_1}{2n+1}$$ and for every $i \in \{1,2,\ldots , m\}$ , there exists $t \in T$ and $s \in [-n,n]$ , such that $a_i=t+s$ .