We will use that the result holds when $V$ is compact [Gub98, Theorem 7.12]. Note that in [Gub98, §7], $K$ was assumed to be algebraically closed, but the argument for [Gub98, Theorem 7.12] does not use this assumption and so we can use the result over any non-archimedean field. Let $f_{+}\coloneqq\max(f,0)$ and $f_{-}\coloneqq\max(-f,0)$ so that $f=f_{+}-f_{-}$. Hence replacing $f$ by $f_{+}$ or $f_{-}$ we can assume that $f\geq 0$.

We can work separately on the connected components of $V$, hence we may assume that $V$ is connected. As in the proof of Proposition 2.8, we can find a locally finite covering $(T_{i})_{i\in I}$ of $V$ made of compact strictly $K$-analytic domains with $I$ finite or countable. In the following, we assume $I={\mathbb{N}}$. The finite case is similar and easier. Applying a compactness argument to the $T_{i}$’s, we can find $(W_{i})_{i\in{\mathbb{N}}}$ and $(U_{i})_{i\in{\mathbb{N}}}$ two locally finite coverings of $V$ by compact strictly $K$-analytic domains of $V$ such that for all $i\in{\mathbb{N}}$ we have $W_{i}\subset U_{i}^{\circ}$.

Let us now fix $\varepsilon>0$ and let us construct a family of piecewise ${\mathbb{Q}}$-linear functions $(\varphi_{i})_{i\in{\mathbb{N}}}$ such that

1. (i)

   for all $i\in{\mathbb{N}}$, ${\rm supp}(\varphi_{i})\subset U_{i}$ and $\varphi_{i}\geq 0$.

2. (ii)

   for all $n\in{\mathbb{N}}$ we have $f\geq\sum_{i=1}^{n}\varphi_{i}\geq f-\varepsilon$ on $\cup_{i=1}^{n}W_{i}$.

3. (iii)

   $f\geq\sum_{i=1}^{n}\varphi_{i}$ on $V$.

Observe that this will conclude the proof of the proposition since then $\varphi\coloneqq\sum_{i\in{\mathbb{N}}}\varphi_{i}$ is a well defined piecewise ${\mathbb{Q}}$-linear function such that $|f-\varphi|\leq\varepsilon$. The rest of the proof is dedicated to construct inductively a family $(\varphi_{i})_{i\in{\mathbb{N}}}$ satisfying the conditions (i), (ii) and (iii).

Let us consider $n\geq 1$ and let us assume that we are given piecewise ${\mathbb{Q}}$-linear functions $\varphi_{1},\ldots,\varphi_{n}$ satisfying the above conditions. We will now construct a piecewise ${\mathbb{Q}}$-linear function $\varphi_{n+1}$ such that $\varphi_{1},\ldots,\varphi_{n+1}$ satisfies the conditions (i), (ii) and (iii).

By the density result in the compact case [Gub98, Theorem 7.12], we know that there exists a piecewise ${\mathbb{Q}}$-linear function $g\colon W_{n+1}\to{\mathbb{R}}$ such that

(2.13.1)

$$f-\sum_{i=1}^{n}\varphi_{i}-\varepsilon\leq g\leq f-\sum_{i=1}^{n}\varphi_{i}\hskip 20.0pt{\rm on}\ W_{n+1}$$

Then by Lemma 2.11 applied to $g$ and $W_{n+1}\subset U_{n+1}\subset V$, there exists a piecewise ${\mathbb{Q}}$-linear function $\Psi\colon V\to{\mathbb{R}}$ which extends $g$ and with ${\rm supp}(\Psi)\subset U_{n+1}$. Then (2.13.1) becomes

(2.13.2)

$$f-\varepsilon\leq\Psi+\sum_{i=1}^{n}\varphi_{i}\leq f\hskip 20.0pt{\rm on}\ W_{n+1}.$$

Then we set

$$\psi\coloneqq\max(0,\Psi).$$

From this definition, we get that ${\rm supp}(\psi)\subset{\rm supp}(\Psi)\subset U_{n+1}$. It is a piecewise ${\mathbb{Q}}$-linear function by Proposition 2.10 (d) and it satisfies $\psi\geq 0$. Now, (2.13.2) combined with the condition (iii) for $n$ yields

(2.13.3)

$$\psi+\sum_{i=1}^{n}\varphi_{i}\leq f\hskip 20.0pt\ {\rm on}\ W_{n+1}.$$

Also, since $\Psi\leq\psi$, we deduce from (2.13.2) that

(2.13.4)

$$f-\varepsilon\leq\psi+\sum_{i=1}^{n}\varphi_{i}\hskip 20.0pt{\rm on}\ W_{n+1}.$$

On the other hand, since $\psi\geq 0$, the condition (ii) for $n$ yields

(2.13.5)

$$f-\varepsilon\leq\psi+\sum_{i=1}^{n}\varphi_{i}\ \hskip 20.0pt\ {\rm on}\ \bigcup_{i=1}^{n}W_{i}.$$

From (2.13.2), (2.13.3), (2.13.4) and (2.13.5), we deduce that

(2.13.6)

$$f-\varepsilon\leq\psi+\sum_{i=1}^{n}\varphi_{i}\leq f\hskip 20.0pt\ {\rm on}\ \bigcup_{i=1}^{n+1}W_{i}.$$

Lemma 2.12 applied to the non negative function $f-\sum_{i=1}^{n}\varphi_{i}\colon V\to{\mathbb{R}}$ and to the compact $K$-analytic domain $\cup_{i=1}^{n+1}U_{i}$ yields a piecewise ${\mathbb{Q}}$-linear function $\chi\colon V\to{\mathbb{R}}$ such that $\chi\geq 0$ and

(2.13.7)

$$f-\sum_{i=1}^{n}\varphi_{i}-\varepsilon\leq\chi\leq f-\sum_{i=1}^{n}\varphi_{i}\hskip 20.0pt\ {\rm on}\ \bigcup_{i=1}^{n+1}U_{i}.$$

We then set

$$\varphi_{n+1}\coloneqq\min(\psi,\chi).$$

By Proposition 2.10 (d), $\varphi_{n+1}$ is a piecewise ${\mathbb{Q}}$-linear function. Since $\psi\geq 0$ and $\chi\geq 0$ we get that $\varphi_{n+1}\geq 0$ and we also get that for $x\in V$, $\psi(x)=0\Rightarrow\varphi_{n+1}(x)=0$. This implies that ${\rm supp}(\varphi_{n+1})\subset{\rm supp}(\psi)\subset U_{n+1}$. Hence (i) is satisfied for $\varphi_{n+1}$.

Let us now prove that

(2.13.8)

$$\sum_{i=1}^{n+1}\varphi_{i}\leq f\ \ {\rm on}\ V.$$

Let $x\in V$. We first suppose that $x\in U_{n+1}$. Then by (2.13.7), we have $\chi(x)+\sum_{i=1}^{n}\varphi_{i}(x)\leq f(x)$. By definition of $\varphi_{n+1}$, we have $\varphi_{n+1}\leq\chi$ hence

$$\varphi_{n+1}(x)+\sum_{i=1}^{n}\varphi_{i}(x)\leq\chi(x)+\sum_{i=1}^{n}\varphi_{i}(x)\leq f(x).$$

If $x\notin U_{n+1}$, then we have $\psi(x)=0$ since ${\rm supp}(\psi)\subset U_{n+1}$, hence $\varphi_{n+1}(x)=0$. So by the condition (iii) for $n$, we get

$$\sum_{i=1}^{n+1}\varphi_{i}(x)=\sum_{i=1}^{n}\varphi_{i}(x)\leq f(x).$$

This proves (2.13.8), whence condition (iii) holds for $n+1$.

Let us finally prove that

$$f-\varepsilon\leq\sum_{i=1}^{n+1}\varphi_{i}\leq f\hskip 30.0pt{\rm on}\ \bigcup_{i=1}^{n+1}W_{i}.$$

The right inequality has been proven in (2.13.8) so it only remains to prove the left inequality. By (2.13.6), we have

(2.13.9)

$$f-\varepsilon\leq\psi+\sum_{i=1}^{n}\varphi_{i}\hskip 30.0pt{\rm on}\ \bigcup_{i=1}^{n+1}W_{i}$$

and by construction (see (2.13.7) having in mind that $W_{i}\subset U_{i}$), we have

(2.13.10)

$$f-\varepsilon\leq\chi+\sum_{i=1}^{n}\varphi_{i}\hskip 30.0pt{\rm on}\ \bigcup_{i=1}^{n+1}W_{i}.$$

Hence (2.13.9) and (2.13.10) yield that

$$f-\varepsilon\leq\min(\psi,\chi)+\sum_{i=1}^{n}\varphi_{i}=\sum_{i=1}^{n+1}\varphi_{i}\hskip 20.0pt{\rm on}\ \bigcup_{i=1}^{n+1}W_{i}$$

which proves condition (ii) for $\varphi_{1},\ldots,\varphi_{n+1}$. By induction, this proves the existence of a family $(\varphi_{i})_{i\in{\mathbb{N}}}$ satisfying conditions (i), (ii) and (iii). ∎