Uniformly equicontinuous sequence of functions that does not converge uniformly

real-analysis functional-analysis uniform-convergence equicontinuity

In a past exam of a course I am taking this semester there is the following question for which I can't find any example by myself. The question is to find an uniformly equicontinuous sequence $(f_n)$ of continuous and bounded functions on $[0,1]$ with real values from which we cannot extract a subsequence that converges uniformly on $[0,1]$.

If any of you had some examples of such sequences I would be really grateful, I just cannot think of one..


Solution 1:

By Arzela-Ascoli, a uniformly bounded, uniformly equicontinuous family of functions on $[0,1]$ always has a uniformly converging subsequence. So one of these assumptions must be violated, and it cannot be the one about equicontinuity.

Consider the sequence $f_n(x) = n$. The functions in this family are uniformly equicontinuous but not uniformly bounded. There is no uniformly convergent subsequence.

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