How could you prove these inequalities wihout induction:($a_k$ are non-negative)
1)$\prod_{k=1}^n(1+a_k)\ge1+\sum_{k=1}^n a_k$
2)$\prod_{k=1}^n(1+a_k)\le1+\frac{\sum_{k=1}^na_k}{1!}+\ldots+\frac{(\sum_{k=1}^na_k)^n}{n!}$
3)$\prod_{k=1}^n(1+a_k)\le\frac1{1-\sum_{k=1}^na_k}, \ \forall\sum_{k=1}^na_k\lt1$
I did not get any positive result by the use of AM-GM inequalities. Induction proves it, but is a little longer for 2). As for 3) the geometric series seems the way. Any hints. Thanks beforehand.