Is the above statement, that if all the terms in a monotonically-decreasing sequence $x_n$ are greater than one, then the sequence $p_n := \prod_{k=1}^{n}x_k$ diverges?
The motivation for this is to know if a sequence coming down from above with a limit of 1 has to have its product diverge...
No. Take your favourite convergent series $\sum_ka_k$ with $a_k>0$ for all $k$ and put $x_k=e^{a_k}$. Then $p_n=\prod_{k=1}^n e^{a_k}=\exp(\sum_{k=1}^n a_k)$ converges, even though all $x_k>1$.