I am trying to prove the following inequality: Given real numbers $a_i\geq0, i=1,2,\ldots n$, then $\prod^n_{i=1}(1+a_i)\geq \sum^n_{i=1} a_i$.
I've tried to do it using induction but I wasn't able to conclude the argument. Is there a simple proof for this? It seems natural, since one of the components of the product will be $\sum_{i=1}^n a_i$ but I am not able to show it rigorously.