We have a sequence of functions $f_n:[0,1]\rightarrow[0,1]$ such that for all $n\in\mathbb{N}$ and $x,y\in[0,1]$, if $|x-y| > \frac{1}{n}$ then $|f_n(x) - f_n(y)| \leq \frac{1}{n}|x-y|$. I need to show that there is a uniformly convergent subsequence.
My idea was to inductively define the sequence something like this:
$n_1 := 1 \\ n_{k+1} := \text{min}\{m>n_k : \text{sup}\{|f_m(x) - f_m(x+\frac{1}{n_k})| : x\in[0,1-\frac{1}{n_k}\}<\frac{1}{n_k}\} $
and then show uniform convergence. I'm not convinced that such a sequence must exist, though.
Any help on showing that it does or suggestions of other strategies would be greatly appreciated.
Hint: The Arzelà-Ascoli Theorem states that every bounded equicontinous sequence of functions in $C^0([a,b], \mathbb{R})$ has a uniformly convergent subsequence.