Assume functions $f_n: [0, \infty) \to \mathbb{R}$ are meeting given conditions:
- $f_n$ are continuous
- for each $n$ there exists limit $\lim_{x\to\infty}f_n(x) = g_n$
- for each $x \in [0, \infty)$, $n \in \mathbb{N}$ $f_n(x) \geq f_{n+1}(x)$
- for each $x \in [0, \infty)$ $\lim_{n\to\infty}f_n(x) = 0$
- $\lim_{n\to\infty}g_n = 0$
Show that sequence $(f_n)_{n\in\mathbb{N}}$ is uniformly convergent on $[0, \infty)$.