This question is related to this other one. I would like to see now if the following holds: $$\prod_{k=1}^n\bigg(1-\frac{a\frac{c}{n}(1-\frac{bc}{n})^{2(n-k)}}{2}\bigg)\stackrel{?}{\to}e^{-\frac{a}{4b}(1-e^{-2bc})}$$ for $a,b,c>0$. From the other question, we know that $$h_n:=\sum_{k=1}^n\frac{c}{n}\bigg(1-\frac{bc}{n}\bigg)^{2(n-k)}\to \frac{1}{2b}(1-e^{-2bc})$$ The limit above feels legit to me because it bears similarity with the sequence converging to $e$, but it may as well be untrue, or true under some conditions. However, I don't know how to evaluate that infinite product because taking logs should be out of the question. Any help super appreciated, especially references on how to evaluate such infinite products with sequences inside (any theorem?).
UPDATE: numerical results point towards the fact that the limit holds, but I am still unable to find a proof.