I am looking for some examples that show that the Arzela Ascoli theorem is "tight". i.e. is there a sequence of functions that is uniformly bounded and equicontinuous on a noncompact set that would not have a uniformly convergent subsequence. Also is there an example of a uniformly bounded non-equicontinuous sequence on a compact set that does not have a convergent subsequence, and similarly by removing the uniformly bounded condition


First question: On $\mathbb R,$ let $f(x) = \sin (\pi x), x= [0,1],$ $f=0$ elsewhere. Define $f_n(x) = f(x-n).$ This is a uniformly bounded, equicontinuous sequence on $\mathbb R$ that converges to $0$ pointwise everywhere, yet fails to have a subsequence that converges uniformly on $\mathbb R.$

Second question. A classic: $f_n(x) = x^n$ on $[0,1].$

Third question: On $[0,1]$ define $f_n(x) = n.$

  • $\begingroup$ My professor also gave a counter example that showed the necessity of compactness which was some sort of triangle function that was zero for all numbers but then between some integers it increased then decreased forming a triangular shape. I didn't understand how his showed that compactness was necessary, can you comment if you are familiar with his example $\endgroup$ – GTOgod Dec 18 '17 at 1:50
  • $\begingroup$ I would need to have more information on that example to comment on it. If you can state it clearly, and you still have a question on it, I'll be happy to comment. $\endgroup$ – zhw. Dec 18 '17 at 21:09

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