IMO 1994 - Shortlisted problems Question 1

A1 (USA) Let $a_0=1994$ and $a_{n+1}=\frac{a_n^2}{a_n+1}$ for each nonnegative integer $n$. Prove that $1994-n$ is the greatest integer less than or equal to $a_n$, $0\le n\le998$.