Equivalence of norms in the space of Lipschitz continuous functions

Question

I know that norms are not necessarily equivalent in $C[0,1]$ because it is an infinite-dimensional space. For example, I was able to show that $||x||_{\infty}$ and $||x||_2$ are not equivalent in $C[0,1]$. However, I need to show that norms are equivalent in the space of Lipschitz continuous functions with common constant $L$ over $[0,1]$, i.e., $\mathrm{Lip}_L[0,1]$. To do this, the problem suggests solving the optimization problem: \begin{align} \mathrm{minimize}\quad&||x||_2 \\ \mathrm{subject\:to}\quad&||x||_{\infty}\geq\epsilon \\ &x\in\mathrm{Lip}_L[0,1]. \end{align} for every $\epsilon>0$. I have two questions. First, how do you show an optimal solution exists? I know Lipschitz functions with a common constant $L$ are equicontinuous, so I am correct to assume that we can use Arzelá-Ascoli's Theorem? Second, how do you go about finding the optimal solution?