By Cauchy-Schwarz inequality,
$$\text{LHS} \le \left( \frac{1}{a_1} + \frac{1}{a_2 - a_1} + \dots + \frac{1}{a_n - a_{n-1}} \right) \left( \frac{a_1}{(1+a_1)^2} + \frac{a_2 - a_1}{(1+a_2)^2} + \dots + \frac{a_n - a_{n-1}}{(1+a_n)^2} \right).$$
Note that $\frac{a_1}{(1+a_1)^2} \le \frac{a_1}{1+a_1}$ and for $i=2, \dots, n$,
$$\frac{a_i - a_{i-1}}{(1+a_i)^2} \le \frac{a_i - a_{i-1}}{(1+a_{i-1})(1+a_i)} = \frac{1}{1+a_{i-1}} - \frac{1}{1+a_i}.$$
Thus, after telescoping,
$$\frac{a_1}{(1+a_1)^2} + \frac{a_2 - a_1}{(1+a_2)^2} + \dots + \frac{a_n - a_{n-1}}{(1+a_n)^2} \le \frac{a_1}{1+a_1} + \frac{1}{1+a_1} - \frac{1}{1+a_n} < 1$$
and we are done.