Exercise 13, § 5

Let $(u_n)$ be a sequence of finite real numbers and let $(\varepsilon_n)$ be a sequence of numbers $\geq 0$, such that $\lim_{n \to \infty} \varepsilon_n = 0$ and $u_{n+1} \geq u_n - \varepsilon_n$ for all integers $n \geq 0$. Let $a = \liminf_{n \to \infty} u_n$ and let $b = \limsup_{n \to \infty} u_n$; show that the set of cluster points of the sequence $(u_n)$ is the interval $[a, b]$.