Proof of Uniform Convergence of continuous functions

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.

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