Exercise 6, § 2

Let $\mathbf{f}$ be a regulated function on the interval ]$a, +\infty$*; suppose that there exists a real decreasing function $g$ on ]$a, +\infty$* such that $\| \mathbf{f}(x) \| \leqslant g(x)$ on this interval and that the integral $\int_a^{+\infty} g(t)\,dt$ converges. Show that the series $\sum_{n=1}^\infty \mathbf{f}(a + nh)$ is absolutely convergent for every $h > 0$, and that