Let $(z\_n)\_{n\in\mathbb N}$ and $(u\_n)\_{n\in\mathbb N}$ be sequences of complex numbers such that: $\lim\_{n\to+\infty}|z\_n|=+\infty$, $\lim\_{n\to+\infty}|u\_n|=+\infty$, $|u\_n|=o(|z\_n|)$ and $(|u\_n|)\_{n\in\mathbb N}$ is non-decreasing. I want to prove that
$$\sum\_{k=0}^{n-1}\ln|z\_n-u\_k|\underset{n\to+\infty}{\sim} n\ln|z\_n|.$$

I did not manage to prove that. Any proof or hint will be welcome. Thanks in advance.