IMC 1997 · Problem 1

Let $\{\varepsilon_n\}_{n=1}^{\infty}$ be a sequence of positive real numbers, such that $\lim\limits_{n \to \infty} \varepsilon_n = 0$. Find

$$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \ln\left( \frac{k}{n} + \varepsilon_n \right),$$

where $\ln$ denotes the natural logarithm.