I'm trying to understand a theorem in a paper on page 14/24. We are given that
$Z = (nq-1) \log \left(\frac{M+nq-1}{nq-1} \right) + M \log \left(\frac{M+nq-1}{M} \right) + \frac{1}{2} \log \left( \frac{M+nq-1}{(nq-1)M} \right) + O(1) .$
Since $e^a > \left(1+\frac{a}{b} \right)^b$ for all $a > 0$ and positive integers $b$, the second term equals
$\log \left(1 + \frac{nq-1}{M} \right)^M < \log e^{nq-1}$
for all positive M and $nq-1 > 0$. This much I'm following, but then it goes on saying:
Since $0 < q < n$ and $n \leq M \leq qn^2$,
$\dfrac{1}{2(nq-1)} \log \left( \frac{M+nq-1}{(nq-1)M} \right)\to 0 \quad \text{for } n \to \infty .$
Where does the factor $\frac{1}{2(nq-1)}$ come from? This might even be trivial, but somehow I'm totally unable to see it.