To prove there exist $M>0$ and $a\_0>0$ such that for $|z|>a\_0$,
$$\left|\frac{p\_n(z)}{q\_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$$ where $p\_n$ and $q\_m$ are the polynomials of degree $n$ and $m$ respectively with $n<m$.

I encountered this question in this [post](https://math.stackexchange.com/questions/2448419/prove-the-integral-int-c-a-fz-eiz-mathrmdz-converges-to-0-when). 

According to the hint, it is enough to show that for large enough $R$ and for every $|z|>R$, we have
$$\left|\frac{z-z\_1}{z-z\_2}\right|<M$$
In fact,
$$\left|\frac{z-z\_1}{z-z\_2}\right|<\frac{|z-z\_2|+|z\_2-z\_1|}{|z-z\_2|}=1+\frac{|z\_2-z\_1|}{|z-z\_2|}<M.$$