I am currently thinking of function-valued random variables. In order to prove a result, I need to approximate by (function-valued) step functions. This naturally leads to the idea of chopping up the function space into finitely many small pieces.
Let $X \subset \mathbb R^d$ be compact and $V$ the set of $1$-Lipschitz functions $X \to \mathbb [0,1]$. Clearly $V$ is equicontinuous and bounded wrt the uniform norm. Since it is also closed it is compact by Arzela-Ascoli. Hence it is totally bounded, meaning:
For any $\epsilon >0$ we can express $V$ as the union of finitely many open sets, each with diameter less than $\epsilon$.
I wonder is there an elementary and not-too-tedious way to prove the above without using the heavy machinery of Arzela-Ascoli? Perhaps we can construct the finitely-many sets directly as balls around some piecewise-defined functions?
The reason I ask is I would like to use the above in the context of optimisation without introducing new terminology (compactness, sequential compactness, equicontinuity et cetera).