Let $(f_n)$ a sequence of continuous function on $[a,b]$ that converges pointwise to $f$. We suppose that $f$ is continuous on $[a,b]$. Prove that the convergence is uniform.

I'm stuck at some point :

Since $|f_n(x)-f(x)|$ is continuous on $[a,b]$, for all $n$, there is $x_n\in [a,b]$ s.t. $$\sup_{[a,b]}|f_n(x)-f(x)|=|f_n(x_n)-f(x_n)|.$$ Since $(x_n)$ is bounded, there is a subsequence that convergence (let denote $x\in [a,b]$ the limit, and still denote the subsequence $(x_n)$). Now $$|f_n(x_n)-f(x_n)|\leq |f_n(x_n)-f_n(x)|+|f_n(x)-f(x_n)|+|f(x_n)-f(x)|.$$ The fact that $|f_n(x)-f(x_n)|\to 0$ and $|f(x_n)-f(x)|\to 0$ is clear. But I can't manage to prove that $|f_n(x_n)-f_n(x)|\to 0$. Any idea ?

  • $\begingroup$ How is the $x$ you use on the triangle inequality defined? You seem to assume $x_n \to x$ but a priori there is no need for $\{x_n\}_n$ to converge. $\endgroup$ – guidoar Mar 30 '19 at 16:00
  • $\begingroup$ @GuidoA.: I updated. $\endgroup$ – user657324 Mar 30 '19 at 16:01
  • $\begingroup$ Okay, that's fair enough, but careful when reindexing: now if we can prove your statement, it will be for a subsequence of $(f_n)_n$. We should still say something for it to hold in the general case. $\endgroup$ – guidoar Mar 30 '19 at 16:02

The statement cannot be proved, since it is false. Suppose, for instance, that you define, for each $n\in\mathbb N$,$$\begin{array}{rccc}f_n\colon&[0,1]&\longrightarrow&\mathbb R\\&x&\mapsto&x^n-x^{2n}.\end{array}$$Then each $f_n$ is continuous and $(f_n)_{n\in\mathbb N}$ converges pointwise to the null function (which is continuous too), but the convergence is not uniform:$$(\forall n\in\mathbb N):f_n\left(\sqrt[n]{\dfrac12}\right)=\frac14.$$

  • $\begingroup$ So the 1st Dini theorem in wikipedia is wrong ? (it's in the french wiki only, I hope you can read it) $\endgroup$ – user657324 Mar 30 '19 at 16:06
  • $\begingroup$ @user657324: You need some form of monotonicity to use Dini's theorem. The chances of a straightforward theorem like this being incorrect are pretty small :-). $\endgroup$ – copper.hat Mar 30 '19 at 16:07
  • $\begingroup$ Sorry, indeed, I didn't see this detail. Thank you. $\endgroup$ – user657324 Mar 30 '19 at 16:08
  • $\begingroup$ Yes, I can read it. I wrote my PhD dissertation in French. And you missed the hypothesis that the sequence $(f_n(x))_{n\in\mathbb N}$ is monotonic. $\endgroup$ – José Carlos Santos Mar 30 '19 at 16:09
  • 1
    $\begingroup$ If you define $f(x)$ as $\lim_{n\to\infty}f_n(x)$, then that's trivial (even without assuming that $(f_n(x))_{n\in\mathbb N}$ increases). The problem lies in proving that the convergence is uniform. $\endgroup$ – José Carlos Santos Mar 30 '19 at 16:28

Let $f=0$ and $f_n$ be the function whose graph is given by the line joining the points $(0,0),({1 \over 2n}, 0)$, $({1 \over n}, 1), ({3 \over 2n}, 0), (1,0)$. The $f,f_n$ are continuous, and $f_n(x) \to f(x)$ everywhere, but the convergence is not uniform since $f_n({1 \over n})-f({1 \over n}) = 1$ for all $n$.


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