Hint: For any pair $x,y\in U$ cover the compact set $\{ty+(1-t)x: t\in [0,1]\}\subset U$ with finitely many open balls $B_1,\ldots,B_n$ such that $f$ is Lipschitz on each $B_i$. Let $X=\{t: t\in \text{ more than one } B_i\}$ which is a finite union of intervals, and let $\epsilon$ be the sum of their lengths. Note that $\epsilon$ can be made arbitrarily small. Apply the triangle inequality, and let $\epsilon\to 0$.
Remark: This actually works for any path-connected $U$, by replacing the set $\{ty+(1-t)x: t\in [0,1]\}$ with $\{p(t):t\in [0,1]\}$ for some path $p$ connecting $x$ and $y$.