I haven't done uniform convergence since last year when I took analysis, and now I have this problem for topology (we're studying metric spaces right now) and I can't remember how to disprove uniform convergence:(adsbygoogle = window.adsbygoogle || []).push({});

f_n: [0,1] -> R , f_n(x)=x^n

Show the sequence f_n(x) converges for all x in [0,1] but that the sequence does not converge uniformily.

I think I have a pretty good concept of uniform convergence, that is, a sequence of functions is uniformily convergent on some domain if the closeness of the function values (in the range) does not depend on the closeness of the values (in thedomain). To disprove uniform convergence, should I show that there exists some epsilon such that |f_n(x) - f_n(y)| >/= epsilon? But I don't know how I would do that.

Thanks for the help!

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# Uniform Convergence

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