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I know that for a finite number of intervals, I can always take the maximum $N$ in each interval, and then my function sequence $f_n(x)$ will uniformly converge in all of the intervals.

but when I have an infinite number of intervals - I know that this is not true, but I am looking for a counter example, I though of $\frac{1}{1+nx}$ - I can say that it uniformly converge in every closed interval inside - like $[1/2,1]$ $[1/4,1/2]$ $[1/4,1/8]$ ... but not on the whole interval $[0,1]$. but something feels "off" with that, I am not sure if thats true

big thanks

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    hmm good to know that! thank you...always helpful2012-03-27

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I would suggest looking for a union of infinitely many bounded intervals(I assume this is what you mean by "infinite intervals") whose union is not bounded, for example [n,n+1]. The function isn't as important as the collection of intervals you choose.