Suppose $\{f_n\}$ be a sequence of continuous function$f_n:S\to \mathbb{R}$ where $S\subset \mathbb{R}$ and $S$ is compact. Suppose for $\{f_n(x)\}$ monotonic decreasing to zero for any $x\in S$. Is $\{f_n\}$ uniformly converge to $ 0$? I know all the definition of convergence and uniformly convergence and compact but still not sure how to start or prove it
Asked
Active
Viewed 1,310 times
0
-
See also $f_1,f_2,\dots$ continuous on $[0,1]$ s.t $f_1 \geq f_2\geq \cdots$ and $\lim_{n\to\infty}f_n(x)=0$. – Dec 03 '12 at 17:30
-
Hmm. I didn't see that when I searched for Dini's theorem prior to answering. Though I now realize that I would have gotten more relevant hits if I had used quotes. – Harald Hanche-Olsen Dec 03 '12 at 17:33
-
Anyhow, to improve the chances of someone finding this one, I edited the title of the question. – Harald Hanche-Olsen Dec 03 '12 at 17:37