Giving a specific example of a positive sequence increasing to 1 and with its partial products having a positive limit In my real analysis class I was asked this which got me stick:

Is there an example of a sequence of real positive numbers increasing to the limit 1 $ \{ a_n \}_{n=1}^{\infty} $ such that the partial products $ a_1 , a_1a_2,a_1a_2a_3,... $ converges to a positive limit?

I thought about it and thought it might be true because I coud not disprove it generally but I cannot come up with an example. Woud someone please be able to provide an example if any? Thanks to all helpers.
 A: $$
\begin{align}
\prod_{k=1}^n\left(1-\frac1{(k+1)^2}\right)
&=\prod_{k=1}^n\frac{k(k+2)}{(k+1)^2}\\
&=\prod_{k=1}^n\frac{k}{k+1}\prod_{k=1}^n\frac{k+2}{k+1}\\
&=\frac1{n+1}\frac{n+2}2\\[3pt]
&\to\frac12
\end{align}
$$
A: Another example is given by $a_n = 1/(1+2^{-2^n})$.
The partial products in this case take the form $a_1...a_k=1/(1+1/4+1/16+\ldots+1/4^{2^k-1})$, converging to $1/\sum_{k=0}^\infty(1/4)^k = 3/4$.
A: Solve this recursively (think through it). Say you want the product to be at least 0.5. So try $a_1 = 0.9$. How big does $a_2$ have to be now to keep the product about $0.5$? Pick an $a_2$ that works say $(1+0.5/0.9)/2$. Now how big does $a_3$ have to be? And so forth.
This process actually gives you an inductively defined sequence that solves your problem, but I'm sure (because I saw the other answers) you can find one with an explicit form (maybe by bounding the one you produce by one with an explicit form).
A: 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 $$a_1 a_2 \dots a_n = \frac{x_{n+1}}{x_1}\to \frac{x}{x_1}>0$$
A: 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
$$
\lim_{N\to \infty}\prod_{n=1}^N\left(1-\frac{x^2}{n^2}\right)=\frac{\sin \pi  x}{\pi  x}.
$$
