$(n_k)$ is a sequence of denominators for the sequence of prinicpal convergents of some irrational number, so $n_k \rightarrow \infty,\delta>0$. Let $0<\varepsilon \ll \delta$. I'm also given that $n_k \le (M+1)^k$.
I have deduced that for any specified natural number $k_0$ $(*)1<\,\frac{\log n_k}{(1+\delta)^{k-k_0}\log n_{k_0}}<1+\frac{\log7}{\delta \log n_{k_0}}.$
Therefore, by taking $k_0$ large I can have the expression in the middle tend to $1$. I'm assuming it is ok to say $ (**)\,\log n_k=A(1+\delta)^{k-k_0}\log n_{k_0},$
as long as I acknowledge that $A$ depends on $k, k_0,$ and $\delta$, where the latter two are predetermined.
$A\rightarrow 0$ since $\lim_{k\rightarrow \infty}A=\lim_{k \rightarrow \infty}\frac{\log n_k}{(1+\delta)^{k-k_0}\log n_{k_0}}\le\lim_{k \rightarrow \infty}\frac{k\log(M+1)}{(1+\delta)^{k-k_0}\log n_{k_0}}=0$
$A\rightarrow 1$ since $(*)$ implies that the ratio tends to $1$ as $k_0$ increases.
Where did I go wrong? I only want $A\rightarrow 0.$