Proof of Inequality involving sum and product without induction

Question

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.

Answer 1

1) $$\prod_{k=1}^n(1+a_k)=1+\sum_{k=1}^nx_k+\sum_{1\leq i<j\leq n}a_ia_j+...\geq1+\sum_{k=1}^na_k$$ 2) follows from Maclaurin's inequality 3) follows from 2) immediately:

Let $a_1+a_2+...+a_n=x$...