I am trying to find an upper bound for the following sequence:
$$(1-p_1)(1-(p_1+p_2))\cdots(1-(p_1+\cdots+p_n))$$
with $n$ groups to multiply. I have written it like this:
$$\prod_{i=1}^n \left({1 - \sum_{j=1}^i {p_j}}\right)$$
Where $p_i$ denotes a constant in the open interval $(0,1)$ and $\sum_{i=1}^n p_i = 1$.
Here's what I've attempted to do:
$$\prod_{i=1}^n \left({1 - \sum_{j=1}^{i}{p_j}}\right) = \prod_{i=1}^n \left(1 - O(1) \right) = \prod_{i=1}^n 1 - \prod_{i=1}^n O(1) = 1 - \prod_{i=1}^n O(1)$$
But I don't know how to bound $1 - \prod_{i=1}^n O(1)$.