Sequence
2007

Problem - 2339
Sequence $\{a_{n}\}$ is defined by $a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $n \ge 1.$ Prove that $[a_{n}] =2007-n$ for $0 \le n \le 1004,$ where $[x]$ denotes the largest integer no larger than $x.$