TITLE: Giving a specific example of a positive sequence increasing to 1 and with its partial products having a positive limit
QUESTION [6 upvotes]: 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.

REPLY [14 votes]: 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 $$a_1 a_2 \dots a_n = \frac{x_{n+1}}{x_1}\to \frac{x}{x_1}>0$$