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"Let (f_n) be an increasing sequence of continuous functions on R. Suppose [tex]\forall x\in\mathbb{R}(f(x)=\lim_{n\rightarrow\infty}f_n(x))[/tex], and suppose that [tex]f(x)<\infty[/tex] for all x, prove that [tex]\{x\in\mathbb{R}:f(x)>a\}[/tex] is open for all a in R."
I think an additional condition of uniform convergence is required.
I think an additional condition of uniform convergence is required.