Exponential type inequality

I found this problem in a set of old analysis prelims. I am looking for some hints after having spent some time working on the problem.

Suppose $a_1,\ldots,a_n$ are real numbers such that $1+a_j>0$ for all $1\leq j\leq n$.If $\sigma_n=\frac{a_1}{1+a_1}+\ldots +\frac{a_n}{1+a_n}\geq0$, then$$1+\frac{\sigma_n}{1!}+\ldots + \frac{\sigma^n_n}{n!}\leq \prod^n_{j=1}(1+a_j)$$with equality only if $a_j=0$ form all $j$.

I tried to apply common inequalities such as $1+x\leq e^x$ but the problem is still elusive. A hint or a solution would be welcome.