Exercise 6, § 4

Let $(u_n)$ be a series of reals $\neq 0$ such that $\lim_{n \to \infty} u_n = 0$. Show that if there exists a number $r$ such that $0 < r < 1$ and if from some point on one has $-1 \leqslant u_{n+1}/u_n \leqslant r$, then the series $(u_n)$ converges.