MathNet 1999 — Part 01 Question 111

The real positive numbers $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ satisfy the relation $a_{n+1}^{2}=a_{n}+1$ for all $n=1,2, \ldots$ Prove that at least one of the $a_{i}$'s must be an irrational number.