Given two numbers $x\in (0,1)$ and $y\in (0,1)$, think of the expression $\min_{n\ge 1} \frac{1-x}{(1-xy)y^{n-1}(1-x^n)}$.
Does the supremum of this expression, namely, $\sup_{x,y\in (0,1)}\min_{n\ge 1} \frac{1-x}{(1-xy)y^{n-1}(1-x^n)}$ be finite or infinite?
I greatly appreciate your help!! I would like to prove this my mechanical engineering research :)