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Upper bound problem in real analysis

  1. Apr 7, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex]\mathcal{F} \subset C(\mathbb{R})[/tex] be a set of continuous
    functions such that for each [tex]x \in \mathbb{R}[/tex] there is an [tex]M_x >
    0[/tex] such that [tex]|f(x)| \leq M_x[/tex] for all [tex]f \in \mathcal{F}[/tex].

    2. Relevant equations

    Prove that there is a nonempty open subset [tex]Y \subseteq X[/tex] and an [tex]M
    > 0[/tex] such that
    |f(x)| \leq M \text{ for all } x \in Y \text{ and all } f \in
    \mathcal{F} \text{.}

    3. The attempt at a solution

    I can't see the point of such a proof as isn't for any [tex]Y \subseteq X[/tex] such an [tex]M = \sup_{x \in Y} M_x[/tex]? What exactly needs to be proven?
  2. jcsd
  3. Apr 7, 2010 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    The supremum could be infinity
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