Integrals I: Feynman's Trick · Superchallenge problem

Let $a_1 = 1$ and $a_{n+1} = a_n + \dfrac{1}{a_n}$. It can be proved by induction that $a_n^2 \ge 2n - 1$ for all $n$. Using that bound, give the smallest integer $N$ for which the inequality guarantees $a_N \ge 10$.