Irrationality of an infinite product

Question

Erdos proved that for a sequence of integers $\{{a}_{n}{{\}}^{\infty }}_{n=1}$ with $\lim_{n\to\infty}a_n^{2^{-n}}=\infty$ the infinite product $$\prod _{n=1}^{\infty }\left(1+\frac{1}{{a}_{n}}\right)$$ converges to an irrational number. See this paper. My question is, what if instead we have a sequence of numbers $b_n$ such that each number in b is irrational? Would the result still be irrational or would it be rational, or do we not know?