I would like to get some kind of bound with say $s\le ...$ from this inequality:
$2^c-2^{c-s}<\left(2^{-s}+1\right){}^{r-1}, c\ge 1,r\ge1,s\ge1, c,r,s\in \mathbb{Z}$
I found the Weierstrass product inequalities but they require $(r-1)2^{-s}\le1$ for the upper bound of $\left(2^{-s}+1\right){}^{r-1}$. How is a problem like this usually tackled?