Let $(x_n)$ be a monotonically decreasing sequence of positive real numbers that is also summable.

Let $(y_n)$ be a sequence of positive real numbers such that $\sum_n x_n y_n$ converges.

Let $(z_n)$ be a monotonically increasing sequence of positive real numbers such that $\sum_n x_n z_n =\infty.$

Assume that the sequences $y_n$ and $z_n$ are such that $2^{-\varepsilon y_n}$ and $2^{-\varepsilon z_n}$ are summable for every $\varepsilon>0.$ Does it follow that there is some $\delta>0$ such that

$$ \sum_n \Big(2^{-\varepsilon y_n}-2^{-\varepsilon z_n}\Big) \ge 0 \text{ for all } \varepsilon \in (0,\delta)?$$

The motivation for this statement to be true is that $z_n$ should be larger most of the time than $y_n$ and we capture this most of the time by taking $\varepsilon$ small.

Please let me know if you have any comments, questions or remarks.