Let $N$ be a finite subset of $\mathbb{N}$, where $|N|>1$. For $i\in\mathbb{N}$, let $a_i$ and $b_i$ be chosen uniformly at random from $N$. Is it true that $\mathbb{P}[\sup_{n\in\mathbb{N}}\{\frac{a_1\cdots a_n}{b_1\cdots b_n}\}=\infty]=1$? Note that given the symmetry involved, I believe this should be the same as asking if $\mathbb{P}[\inf_{n\in\mathbb{N}}\{\frac{a_1\cdots a_n}{b_1\cdots b_n}\}=0]=1$.

This question came up while looking at the coarse geometry of trees, and if it is true then it has a nice corollary in that area. Unfortunately I do not have the background in probability to tell if this is an easy question or not, let alone answer it. I suspect it is true based only on the intuition of the people I have already asked, and doing some tests in python.