Lower bound on $\prod_{n=1}^{N} (1 + a_n)$ when $|a_n| = O(1/n)$

Question

I found in a paper the inequality

\begin{equation}\prod_{n=1}^{N} (1 + a_n) \geq \exp\left(-O\!\left(\left|\sum\nolimits_{n=1}^{N} a_n\right| +1\right)\right)\end{equation}

where we know that

$$-\frac{1}{2} < a_n < \frac{1}{2} \; \text{ and } \; a_n = O\!\left(\frac{1}{n}\right).$$

I tried to apply $\log(\exp(.))$ and some basic inequalities on the logarithm, but couldn't find a correct argumentation. Can someone help me?