[Math] Arzela Ascoli counterexamples

Question

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

Answer 1

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 $f_n(x) = n.$