Since piecewise ${\mathbb{Q}}$-linear functions are dense in the compact case [Gub98, Theorem 7.12], there exists a piecewise ${\mathbb{Q}}$-linear function $g\colon W\to{\mathbb{R}}$ such that $f-\varepsilon\leq g\leq f$ on $W$. Since $W$ is compact, there is a non-zero $k\in{\mathbb{N}}$ such that $kg$ is piecewise linear. By Proposition 2.6 and Proposition 2.8 applied to the formal metric on ${\mathcal{O}}_{V}$ associated to $kg$, there exists a piecewise ${\mathbb{Q}}$-linear function $\psi\colon V\to{\mathbb{R}}$ which extends $g$. We then set $\varphi\coloneqq\max(\psi,0)$. By Proposition 2.10 (d), $\varphi$ is piecewise ${\mathbb{Q}}$-linear. By definition, we have $\varphi\geq 0$. We have $\psi\leq f$ on $W$ and $f$ is non-negative, hence we have $\varphi\leq f$ on $W$. Finally, since $f-\varepsilon\leq\psi$ on $W$ we also have that $f-\varepsilon\leq\max(\psi,0)=\varphi$ on $W$. ∎