Suppose K is compact and ${f_n}$ is a sequence of continuous functions on K which converges pointwisely to a continuous function f(x), and $f_n(x)\geq f_{n+1}(x)$ for all x $\in$ K and n $\in$ N. Show $f_n \rightarrow f$ uniformly.

My thoughts:

To prove uniform convergence I think we should use the definition(epsilon delta). But I'm not sure how to use other conditions. I was trying to combine "uniform continuous" and "pointwise convergence" but it didn't work.

  • $\begingroup$ $f_n(x)\ge f_{n+1}(x)$ may not be enough, since $f_n(x)\to -\infty$ is not ruled out. $\endgroup$ Oct 30 '19 at 2:46
  • $\begingroup$ math.stackexchange.com/questions/2541875/… $\endgroup$
    – user284331
    Oct 30 '19 at 2:47
  • $\begingroup$ @herbsteinberg $f_n \to f$, and $f$ is continuous on $K$, so that will not happen. $\endgroup$
    – xbh
    Oct 30 '19 at 2:49
  • $\begingroup$ Example: $K=[0,1]$ and $f_n(x)$ defined as follows $f_n(x)=\frac{-1}{x}$ for $x\gt \frac{1}{n}$ and $f_n(x)=-n$ for $0\le x\lt \frac{1}{n}$. $\endgroup$ Oct 30 '19 at 2:59
  • $\begingroup$ Then $f(0) = -\infty$, so $f$ is not continuous at $0$. $\endgroup$
    – xbh
    Oct 30 '19 at 3:06

Actually, this is the Dini’s Theorem. You can see this lecture note: http://www.math.ubc.ca/~feldman/m321/dini.pdf

  • $\begingroup$ Thank you so much! $\endgroup$
    – Matherine
    Oct 30 '19 at 14:08

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