Pointwise bounded, uniformly continuous family of functions locally uniformly bounded

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Homework Statement


Let f_n:[a,b] -> R be a pointwise bounded, continuous family. Prove there exists an interval (c,d) < [a,b] on which f_n is uniformly bounded.


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The Attempt at a Solution


I'm stuck. If we have equicontinuity, then this is easy, so i'm thinking we need to prove we have some kind of local equicontinuity or equicontinuity at one point, but I've not been successful in breaking through with these ideas. I don't have a hook!
 

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