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For each integer [itex]n\geq 1[/itex], let [itex]f^{(n)}:(-1,1) \to \mathbb{R}[/itex] be the function [itex]f^{(n)}(x):= x^n[/itex]. Prove that [itex]f^{(n)}[/itex] converges pointwise to the zero function, but does not converge uniformly to any function [itex]f:\mathbb{R} \to \mathbb{R}[/itex].

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I already proved that it converges pointwise to the zero function, I am having trouble with the second part.

If [itex]f^{(n)}[/itex] is going to converge uniformly to any function, then it has to converge uniformly to the zero function (f(x)=0 for all x), since we know that it converges pointwise to the zero function.

So we want to show that it does not converge uniformly to the zero function. That is, we want to show that there exists [itex]\epsilon > 0[/itex] such that for all [itex]N > 0[/itex] there is an [itex]n>N[/itex] and [itex]x \in (-1,1)[/itex] such that [itex]|x^n| > \epsilon[/itex].

I first tried [itex]\epsilon = 1/5, \ x =\frac{N}{N+1}, \ n = N+1[/itex]

If [itex]x^n[/itex] is increasing then we are done, but I can't quite prove it. Though I do think it is true.

Any ideas on how to prove that [itex]x^n[/itex] is increasing. Or of a better example? Thanks!

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Definitions:

Uniform Convergence: Let [itex](f^{(n)})_{n=1}^{\infty}[/itex] be a sequence of functions from one metric space [itex](X,d_X)[/itex] to another [itex](Y,d_Y)[/itex], and let [itex]f:X\to Y[/itex] be another function. We say that [itex](f^{(n)})_{n=1}^{\infty}[/itex] converges uniformly to [itex]f[/itex] on [itex]X[/itex] if for every [itex]\epsilon > 0[/itex] there exists [itex]N>0[/itex] such that [itex]d_Y(f^{(n)}(x),f(x))< \epsilon[/itex] for every [itex]n>N[/itex] and [itex]x\in X.[/itex]

Pointwise Convergence: Let [itex](f^{(n)})_{n=1}^{\infty}[/itex] be a sequence of functions from one metric space [itex](X,d_X)[/itex] to another [itex](Y,d_Y)[/itex], and let [itex]f:X\to Y[/itex] be another function. We say that [itex](f^{(n)})_{n=1}^{\infty}[/itex] converges uniformly to [itex]f[/itex] on [itex]X[/itex] if for every [itex]x[/itex] and [itex]\epsilon>0[/itex] there exists [itex]N>0[/itex] such that [itex]d_Y(f^{(n)}(x),f(x))< \epsilon[/itex] for every [itex]n>N[/itex].

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# Homework Help: Uniform convergence

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