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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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Your example is fine. It's not uniformly convergent on $(0,1]$ but it is uniformly convergent on any interval of the form $[\epsilon,\delta]\subset(0,1]$. – David Mitra Mar 27 '12 at 18:21
hmm good to know that! thank you...always helpful – YNWA Mar 27 '12 at 20:12

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.

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thank you for your comment. ! :) – YNWA Mar 27 '12 at 20:12

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