I'm working on a proof and there's a part I don't get. $K$ is a compact subspace of the space of $2{\pi}$-periodical real valued integrable functions, and for every integer $n$, $c_n$ is the application that maps a funtion to its $n$-th fourier coefficient.
Now the sequence $(c_n)_n$ converges simply to the null function in $K$, and the $c_n$ are all $1$-Lipchitz (where $c_n$ is viewed as an application from $L^1[0,2{\pi}]$ to $\Bbb{R}$), so now comes the part I don't get: the Dini theorem "Lichitz version" is invoked to prove that the convergence in uniform in $K$.
I looked for the Dini theorem "Lichitz version" but all I could find were the regular Dini theorems, and nothing involving Lipchitz functions (or non monotone functions for that matter)
It sounds like they are using the following principle:
Proof: We're given $\epsilon>0$. There is a finite $(\epsilon/2L)$-net $X$ of $K$ (i.e. $\bigcup_{x\in X}B(x,\epsilon/2L)$ covers $K$). Since $X$ is finite, there exists some $N$ such that $|c_n(x)|\leq \epsilon/2$ for all $n\geq N$ and $x\in X.$ Every point $k\in K$ satisfies $d(k,x)\leq \epsilon/2L$ for some $x\in X,$ and the Lipschitz property gives $|c_n(k)-c_n(x)|\leq \epsilon/2.$ So $|c_n(k)|\leq\epsilon$ for all $n\geq N$ and $k\in K,$ as required for uniform convergence.
A more general (and better-known) statement is the Arzelà–Ascoli theorem. The main difference is that the Lipschitz property can be weakened to just equicontinuity.