ILL 1983

Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$ . Suppose that $$\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.$$ For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$ , prove that $u_n \to +\infty$ as $n \to +\infty.$