Proof of inequality $\prod_{k=1}^n(1+a_k) \geq 1 + \sum_{k=1}^n a_k$ with induction I have to show that $\prod_{k=1}^n(1+a_k) \geq 1 + \sum_{k=1}^n a_k$ is valid for all $1 \leq k \leq n$ using the fact that $a_k \geq 0$.
Showing that it works for $n=0$ was easy enough. Then I tried $n+1$ and get to:
$$\begin{align*}
\prod_{k=1}^{n+1}(1+a_k) &= \prod_{k=1}^{n}(1+a_k)(1+a_{n+1}) \\
&\geq (1+\sum_{k=1}^n a_k)(1+a_{n+1}) \\
&= 1+\sum_{k=1}^{n+1} a_k + a_{n+1}\sum_{k=1}^n a_k
\end{align*}$$
In order to finish it, I need to get rid of the $+ a_{n+1}\sum_{k=1}^n a_k$ term. How do I accomplish that? It seems that this superfluous sum is always positive, making this not really trivial, i. e. saying that it is even less if one omits that term and therefore still (or even more so) satisfies the $\geq$ …

We want to show :
$$\left(\frac{1}{a_{n+1}}+1\right)\prod_{i=1}^{n}\left(1+a_{i}\right)>1+\frac{1}{a_{n+1}}+\sum_{i=1}^{n}\frac{a_{i}}{a_{n+1}}$$
We introduce the function :
$$f(a_{n+1})=\left(\frac{1}{a_{n+1}}+1\right)\prod_{i=1}^{n}\left(1+a_{i}\right)-1+\frac{1}{a_{n+1}}+\sum_{i=1}^{n}\frac{a_{i}}{a_{n+1}}$$
If we differentiate and multiply by $a_{n+1}^2$ we fall on the hypothesis induction . Moreover we deduce that the function is decreasing so we have :
$$f(a_{n+1})\geq \lim_{a_{n+1}\to \infty}f(a_{n+1})=\operatorname{induction hypothesis}\geq 0$$
Remins to multiply by $a_{n+1}$ and conclude .
The advantage is : we get a $n$ refinements and we know much more on the behavior of the difference .
