What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?


Only 2 results I have found so far are,


* That the $\infty$-norm covering number for $L$-Lipschitz functions constrained to map $[0,1]^d \rightarrow [0,1]$ is $\exp\left(\Theta\left( L/\epsilon\right)^d\right)$. And for this I could not find a reference for the proof.
* Another such $\infty$-norm covering number count for $1$-Lipschitz functions mapping an unit diameter metric space to $[-1,1]$ was given in this previously unanswered question here, [metric entropy for Lipschitz functions](https://mathoverflow.net/questions/154427/metric-entropy-for-lipschitz-functions)


Even here I can't find the proof anywhere and the link in the MO question seems to go to something else (and the original reference in the Uspekhi Mat. Nauk, 1959, Volume 14, Issue 2(86), Pages 3–86, is in Russian and hence I can't read it).


Is there some reason why we don't have (is provably impossible?) such counts when either the range or the domain of the function space is unbounded? 


It would be great if someone can maybe reference the proofs for the two results quoted here! 



Covering number of Lipschitz functions
Here is a reference to a more general result: Lipschitz functions over a doubling metric space (rather than $[0,1]^d$). The $||\cdot||\_\infty$ $\epsilon$-metric entropy of such functions is, disregarding log factors, of order $(D/\epsilon)^{ddim}$, where $D$ and $ddim$ are the diameter and doubling dimension of the metric space, respectively. A modern proof may be found in Lemma 4.2 here:
<https://www.sciencedirect.com/science/article/pii/S0304397515009469>


Edit: As pointed out in the comments, the result above is stated without proof. However, a proof is given in Lemma 6 here:
<https://ieeexplore.ieee.org/document/7944658/>


