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Open Set 2

  1. Apr 9, 2006 #1
    "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.
  2. jcsd
  3. Apr 9, 2006 #2


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    No, you don't need uniform convergence. The proof is essentially a one- or two-liner.
  4. Apr 9, 2006 #3
    I got it. Thanks.
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