# A question about counter example of sequence of functions

(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 ?

• What is your question? – José Carlos Santos Mar 17 '18 at 10:30
• @JoséCarlosSantos..is my example satisfies all the above properites – 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).