I am currently working through understanding and replicating a proof in a paper, and I am struggling to understand a key step towards the end of the proof. The paper is on SIS Epidemic models on Networks but this proof is based on theory relating to the solution to Gambler's Ruin.
I want to show that $$ \frac{1-r}{1-r^m}(\frac{1-r^{m-1}}{1-r^m})^{\lfloor r^{-m+1}\rfloor} \geq \frac{1-r}{e}(1+\mathcal{O}(r^m))$$,
where $r$ is a positive constant strictly less than $1$, and $m$ is some positive integer (could be tending to infinity). I am noticing where the exponential could come into some terms, but they keep cancelling out and I never manage to get anything that looks like this form. Any help would be greatly appreciated!