Exercise 6, § 4

Let $(u_n)$ be a series with real terms $\neq 0$, such that $\lim_{n \to \infty} u_n = 0$. Now prove that, if there exists a number $r$ such that $0 < r < 1$ and that, from a certain value of $n$ onwards, one has $-1 \leq u_{n+1}/u_n \leq r$, the series $(u_n)$ is convergent.