Let the function $f_n$ : $[0,1] \rightarrow [0,1]$ satisfy
\begin{align*}
\vert f_n(x)-f_n(y)\vert \leq \vert x-y \vert \textrm{ whenever } \vert x-y \vert \geq \frac{1}{n}
\end{align*}
Show that the sequence $\{f_n\}$ has a uniformly convergent subsequence.
I try to use the Arzela-Ascoli Theorem. Thus, I check to $\{f_n\}$ is uniformly bounded and equicontinuous. But, I don't know how to prove that when $\vert x – y \vert <\frac{1}{n}$
Any help is appreicated…
Thank you!
Best Answer
Although functions $f_n$ are not equi-continuous (and not even continuous), one can still apply the Arzela-Ascoli diagonalization argument, with a slight preparation.
Let $n\in \mathbb{N}$ be fixed. Take any $x_1\in (0,1)$ and consider the set $$U_{x_1} = (x_1 - \frac 2n, x_1 + \frac 2n) \setminus [x_1-\frac 1n,x_1 + \frac 1n].$$ Then $U_{x_1}$ is an open set, and for any $x\in U_{x_1}$ we have, by the condition of the problem, that $$ | f_n(x) - f_n(x_1) |\leq \frac 2n. $$ Hence, for any $x,y \in [0,1]$ with $|x-x_1|, |y-x_1| \leq 1/n$ we get $$ \tag{1} |f_n(x) - f_n(y)| \leq |f_n(x) - f_n(x_1) | + |f_n(y) - f_n(x_1)| \leq \frac 4n. $$
Let $x_1$ run over rationals $Q$ (say), then $\{(x_1 - 1/n, x_1 + 1/n)\}_{x_1 \in Q}$ will serve as an open cover of $[0,1]$. Extracting a finite sub-cover and applying $(1)$ to pass from one interval of length $2/n$ to its neighbor, we get $$ \tag{2} |f_n(x) - f_n(y) |\leq \max\left\{ \frac 8n, |x-y| \right\}. $$
Now, $(2)$ shows "asymptotic equi-continuity", the oscillations of $f_n$ decay uniformly as $n\to \infty$, and we are now ready to apply the argument in the proof of Arzela-Ascoli using $(2)$ instead of the usual equi-continuity.