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Upper bound problem in real analysis

  • #1

Homework Statement



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].

Homework Equations



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

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?
 

Answers and Replies

  • #2
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
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The supremum could be infinity
 

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