- #1

- 102

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"[itex] (x^n)_{n\in \mathbb{N}} [/itex] converges uniformly to zero on [itex] [-\alpha, \alpha] [/itex] for all [itex]0 < \alpha < 1[/itex] but does not converge uniformly to zero on [itex](-1, 1) [/itex]"

My problem is that isn't this interval [itex] [-\alpha, \alpha] [/itex] for all [itex]0 < \alpha < 1[/itex] and [itex] (-1,1) [/itex] the same thing? Am I missing some key analysis fact?

I keep thinking of the example where the sequence functions given by [itex] f_n(x) = x + \dfrac{1}{n} [/itex] converges uniformly to [itex] f(x) = x [/itex] for all [itex]x \in \mathbb{R} [/itex] and [itex]\mathbb{R}[/itex] is clopen. Do intervals have to be closed and bounded for uniform convergence to work?