Upper bound problem in real analysis

[complexnumber]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

I can't see the point of such a proof as isn't for any Y \subseteq X such an M = \sup_{x \in Y} M_x? What exactly needs to be proven?

 

[Office_Shredder]

The supremum could be infinity