Limit, Riemann Sum, Integration, Natural logarithm

A simpler and more direct choice is to write \begin{align}\lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{mn} \right) &= \lim_{n \to \infty} \sum_{r=n+1}^{mn} \frac{1}{r} \\&= \lim_{n \to \infty} \frac{1}{n} \sum_{r=n+1}^{mn} \frac{1}{r/n} \\&= \int_{x=1}^m \frac{1}{x} \, dx \\&= \log m,\end{align} but your solution is valid.