I was thinking about marriage proposals. If the probability of someone saying yes to the proposal is unvarying, say $p$, all one needs to do to get to a yes is to ask enough times. The probability of getting a no n times is $(1-p)^n$, which will surely converge to zero for any $p>0$.
I started to wonder, what if the successive probability of a 'yes' event decreases with each trial? In this case, the probability of getting a no n times is $\prod_{i=1}^{n} (1-p_i)$ and $p_1>p_2...>p_n$. This would perhaps be a better representation of reality.
What can be said about this series as n goes to infinity, or in general? I know the value must be between 0 and 1, but not much more than that.
Is any monotonic decrease in $p_i$ enough to make this value not approach to 0 as n approaches infinity? Or are there some conditions on the 'rate' of decrease in $p_i$ for this value to not be zero?