An Iterative Estimate

A sequence has $a_0 = \dfrac12$ and $a_n = a_{n-1} + \dfrac{1}{n^2}a_{n-1}^2$.

1. Prove $\dfrac{1}{a_{n-1}} - \dfrac{1}{a_n} < \dfrac{1}{n^2}$. 2. Prove $a_n < n$ for $n \ge 1$. 3. Prove $\dfrac{1}{a_n} \le \dfrac56 + \dfrac{1}{n+1}$ for $n \ge 1$.