Can the product of a sequence of numbers between 0 and 1 converge to positive?

Let $x_n \in (0,1)$, is it possible that $\prod_{n=1}^\infty x_n >0$ ? I think it isn't, because such small numbers multiplied together will become smaller and smaller, but I am not sure if there is a positive lower bound for the product. Thanks!

Solution 1:

For example, set $$a_n = 1+\frac{1}{2^n}$$

and then $$b_n = \frac{a_n}{a_{n-1}} < 1,$$ so that $$\prod_{k=1}^n b_k = \frac{a_n}{2}.$$

Of course, it implies that your $x_n \to 1$. In case $x_n \not\to 1$ it means that there exists $\alpha \in (0,1)$ such that $x_n < \alpha$ infinitely many times, and $\prod_k^n x_k \leq \alpha^{\#\{k \leq n \mid x_k < \alpha\}} \to 0$.

Hope that helps ;-)

Solution 2:

Well, if $\prod_n x_n = \theta$, and $x_n, \theta >0$, then $\ln(\prod_n x_n ) = \sum_n \ln x_n = \ln \theta$. Then you can use your knowledge of summations to find an example.

Here is one: $\theta = e^{-\frac{1}{2}}$, and $x_n = e^{-\frac{1}{2^{n+2}}}$.