Giving a specific example of a positive sequence increasing to 1 and with its partial products having a positive limit

Take any positive decreasing sequence $x_1 , x_2, \dots$ that converges to a positive value $x$
Define $a_n = \frac{x_{n+1}}{x_n}$

Then $$a_1 a_2 \dots a_n = \frac{x_{n+1}}{x_1}\to \frac{x}{x_1}>0$$

One may choose $$ a_n=1-\frac1{4n^2} $$ we have $$ \lim_{n\to \infty}a_n=1 $$ and we have $$ \lim_{N\to \infty}\prod_{n=1}^Na_n=\lim_{N\to \infty}\prod_{n=1}^N\left(1-\frac1{4n^2}\right)=\frac2{\pi}>0 $$ where we have used $$ \lim_{N\to \infty}\prod_{n=1}^N\left(1-\frac{x^2}{n^2}\right)=\frac{\sin \pi x}{\pi x}. $$

$$ \begin{align} \prod_{k=1}^n\left(1-\frac1{(k+1)^2}\right) &=\prod_{k=1}^n\frac{k(k+2)}{(k+1)^2}\\ &=\prod_{k=1}^n\frac{k}{k+1}\prod_{k=1}^n\frac{k+2}{k+1}\\ &=\frac1{n+1}\frac{n+2}2\\[3pt] &\to\frac12 \end{align} $$