so, I want to demonstrate the validity of the following inequality:
$$ \prod_i(1-p_i) \geq 1 - \sum_ip_i $$
with $p_i \in [0,1]$, it is always true, which it seems to be always the case if you test it with some random numbers.
So, i tried to do a demonstration. I started by noticing that I could replace $p_i$ with $1-q_i$ obtaining the following inequality:
$$ \prod_iq_i \geq 1 - \sum_i(1 - q_i) $$
That, if the quantity of numbers is $n$ becomes:
$$ \prod_iq_i \geq 1 - n + \sum_iq_i $$
Now it's possible to establish the truthfulness of the inequality under certain conditions, by noticing the range of the various terms:
$$ \prod_iq_i \in [0,1]\\ \sum_iq_i \in [0,n] $$
Therefore: $$ 1 - n + \sum_iq_i \in [-(n+1),1] $$
Luckily, at least I have now demonstrated that if $\sum_iq_i \leq n-1$, then the inequality is true, which is good, because it covers most cases (especially for $n \rightarrow \infty$)!
Now I am left with the problem of the last case: $\sum_iq_i \in [n-1,n]$, and I am stuck.
How would you demonstrate the inequality for this subset of cases?
A last few passages: if I use a $\delta$ term for the uncovered subset of cases as follows: $$ \sum_i q_i = n-1+\delta $$
Then all I have to demonstrate is that $\prod_iq_i \geq \delta$.