(A) given an example of sequence $f_n:[0,1] \to \mathbb{R}$ such that $\{f_n\}^\infty_{n=1} $ are continuous, uniformly bounded, converging pointwise without having a uniformly convergent subsequence.

(B) Can the $\{f_n\}^\infty_{n=1} $ in the above example be an equicontinuous family?

$f_n:[0,1] \to \mathbb{R}$ defined by $f_n(x)=x^n$, does it satisfy all the above properties ?

  • 2
    $\begingroup$ What is your question? $\endgroup$ – José Carlos Santos Mar 17 '18 at 10:30
  • $\begingroup$ @JoséCarlosSantos..is my example satisfies all the above properites $\endgroup$ – Inverse Problem Mar 17 '18 at 10:55

Yes, your example satisfies all the properties that you mentioned: each $f_n$ is continuous, the sequence is uniformly bounded (the range of each $f_n$ is a subset of $[0,1]$) and it has no uniformly convergent subsequence (such a subsequence would converge uniformly to a discontinuous function, which is impossible).


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