Proving Openness of {f(x)>a} in R

[Treadstone 71]

"Let (f_n) be an increasing sequence of continuous functions on R. Suppose $$\forall x\in\mathbb{R}(f(x)=\lim_{n\rightarrow\infty}f_n(x))$$, and suppose that $$f(x)<\infty$$ for all x, prove that $$\{x\in\mathbb{R}:f(x)>a\}$$ is open for all a in R."

I think an additional condition of uniform convergence is required.

 

[AKG]

No, you don't need uniform convergence. The proof is essentially a one- or two-liner.

 

[Treadstone 71]

I got it. Thanks.