I have the following problem:
Let $\{f_n\}_{n = 1}^\infty$ be a sequence of functions such that, for all $n \in \mathbb{N}$, $f_n:[0,1] \to [0,1]$ is continuous, and for all $x\in [0,1]$, we have that $f_n(x) \to 0$ and $f_{n+1}(x) \le f_n(x)$. Show that $f_n \to 0$ uniformly.
My idea so far is to get for every $n \in \mathbb{N}$ a $c_n \in [0,1]$ such that $f_n(c_n) = \sup_{x \in [0,1]} f_n(c_n)$ and to show that $f_n(c_n) \to 0$. I know that passing to a subsequence I can assume that the $c_n$ converge to some $c \in [0,1]$, and I know that sequence $\{f_n(c_n)\}$ is monotonically decreasing, so it has a limit. My instinct tells me to show that $\lim_{n \to \infty}f_n(c_n) = \lim_{n \to \infty}f_n(c)$, but I haven't been able to do this last step. I feel like I'm close to a solution, but am I really? Does anyone have any idea how to close this proof or of how to do a different proof?