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## 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?