Let $0<U_n<1$ be a sequence of decreasing numbers that converges to $0$. Let us consider $V_n$: $$V_n = \prod_{k=0}^{n} (1-U_k)$$
Since $V_n$ is decreasing and lower bounded by $0$, we can conclude that it converges as $n$ tends to $\infty$. Let us write this limit $V_\infty$. What would be a simple/minimal condition on $U_n$ to guarantee that $V_\infty > 0$ ?
For instance, I already know that, if $U_n$ is upper bounded by a geometric sequence with ratio $r<1$ and initial value $a<1$, then: $$ V_n \geq \prod_{k=0}^{n} (1-ar^n) = (a;r)_n,$$ with $(a;r)_n$ denoting the $q$-Pochhammer symbol. But I'm not even sure on the condition on $a$ and $r$ such that $(a;r)_\infty > 0$. It seems to me that it is always the case, but after some google search I've not found the information anywhere.
