We know the sequence space $c_{o}$, which is the space of all sequences converging to zero, and it is a Banach space. Consider a subset $S$ of $c_{o}$ of (some) such sequences which are decreasing to 0, i.e., $S=\left\{\{s_{n}\}: \{s_n\}\; \text{is decreasing and}\; s_{n}\to 0\right\}$.
Can we find a sequence $\{s^'_{n}\}\in S$ such that $\lim_{n\to\infty}\frac{s^'_{n}}{s_{n}}=0$ for all sequences $\{s_{n}\}\in S$? (other than $s^'_{n}$)
(I've posted a similar question but in another form, but I re-formulate it in a way to be easy to understand, and contains the idea I want. So, since I didn't get answers for the old question it can be deleted)
My previuos guess was to take $s^'_{n}$ to be the product of all sequences in $S$, but I'm not sure if this is correct!
Any help is appreciated!