# Open Set 2

1. Apr 9, 2006

"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.

2. Apr 9, 2006

### AKG

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

3. Apr 9, 2006