TITLE: Does the multiplication of countably infinite many numbers between $0$ and $1$ equal $0$?
QUESTION [9 upvotes]: Suppose every term of a countably infinite sequence $x_1,x_2,\dots$ is between $0$ and $1$, i.e. $0<x_i<1$ for every $i$. Does $\mathop {\lim }\limits_{n \to \infty } \prod\limits_{i = 1}^n {{x_i}} $ equal $0$ or there is counterexample that it might not be $0$? Thank you!

REPLY [3 votes]: Good result to know: Suppose $a_n\in (0,1)$ for all $n.$ Then $\prod_{n=1}^\infty (1-a_n) > 0$ iff $\sum a_n < \infty.$ The proof is a nice exercise in taking logs and using $\log (1+u) = u + o(u).$