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

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 ;-)

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 $x_n = e^{-\frac{1}{2^{n+2}}}$.

Take your favorite converging series with positive general term $$ \sum_{n\geq 1}y_n=S. $$ Then set $$ x_n=e^{-y_n}=\frac{1}{e^{y_n}}. $$ You have $$ \prod_{n\geq 1}x_n=e^{-S}=\frac{1}{e^S}. $$