RIOPLATENSE 1995 Problem 4

Given the natural numbers $a$ and $b$ , with $1 \le a <b$ , prove that there exist natural numbers $n_1<n_2< ...<n_k$ , with $k \le a$ such that $$\frac{a}{b}=\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}$$