Inequality with ceiling function

Question

I was working through one lemma in paper and ran into the following inequality: $$\left\lceil \dfrac{M^n}{nM^2-n+1}\right\rceil \ge \dfrac{M^n}{n(M^2-1)}>\dfrac{M^{n-2}}{n},$$ where $M$ and $n$ are positive integers. The last inequality is obviuos, but the first one seems quite weird. I was trying to use that $x\leq \lceil x\rceil<x+1$ but it does give any results.

Answer 1

$$\Delta:=\frac{M^n}{n(M^2-1)}-\dfrac{M^n}{nM^2-n+1}=\frac{M^n}{n(M^2-1)(n(M^2-1)+1)}>1 $$ for $M$ sufficiently large and $n\geqslant 2$. This appears to be problematic since $$0\leqslant\left\lceil\dfrac{M^n}{nM^2-n+1}\right\rceil-\dfrac{M^n}{nM^2-n+1}<1$$