Let $f_1,f_2,\ldots$ be continuous functions on $[0,1]$ satisfying $f_1 \geq f_2 \geq \cdots$ and such that $\lim_{n\to\infty} f_n(x)=0$ for each $x$. Must the sequence $\left\{f_n\right\}$ converge to $0$ uniformly on $[0,1]$?
I want to say yes and I want to try to invoke Arzela-Ascoli Theorem. My intuition tells me that for large $n$ we have that the slope of $f_n$ on small enough interval has slope less than 1. So $f_n$ must be Lipschitz and hence equicontinuous. So it must have a uniformly convergent subsequence, but that gives us that $f_n$ must converge uniformly to $0$.
Any help would be appreciated.