\(The sequence $(a_n)$ is defined by $a_1 = \frac{1}{2}$ and \[a_n = a_{n - 1}^2 + a_{n - 1}\]for $n \ge 2.$ Prove that \[\frac{1}{a_1 + 1} + \frac{1}{a_2 + 1} + \dots + \frac{1}{a_n + 1} < 2\]for all $n \ge 1.$\)
