f$ and $f_k\equiv f$ on $[2^{-k},+\infty)$. Notice that $f_k$ is continuous, sub-additive and increasing on $[0,+{\infty})$ and that $f_k(0)=0$ with $\lim_{t{\rightarrow}0} \frac{f_k(t)}{t}=+{\infty}$. We define the associated energy for functions $v\in\P(\hat{\Omega})$ as ${\mathcal{E}}_{f_k}(v,\hat {\Omega}):=\int_{J_{v}\cap\hat{\Omega}}f_k([v])\dH^1$.\
Now we note $\P_k(\hat{\Omega})$ the set of functions $v\in \P(\hat{\Omega})$ such that $v(\hat{\Omega})\subset 2^{-k}{\mathbf{Z}}$. For these functions we have $|v^+(x)-v^-(x)|\geq 2^{-k}$ for $\H^1$-almost every $x\in J_v$. Consequently, there holds $${\mathcal{E}}_{f_k}(v)\, =\, {\mathcal{E}}_f(v).$$ For each fixed $k\geq0 $, let us introduce the function $$u_k=2^{-k}\lfloor2^k u\rfloor$$ where $\lfloor t\rfloor$ denotes the integer part of the real $t$. Note that $u_k\in \P_k(\hat{\Omega})$ with $J_{u_k}\subset J_u$ and $\|u-u_k\|_\infty\leq 2^{-k}$. Notice also since ${\left}|(u_k^+-u_k^-)-(u^+-u^-){\right}|\leq 2^{-k}$ we have $$\label{eq:diffuuk}
|Du_k-Du|(\hat{\Omega})\, \leq\, 2^{-k}\H^1(J_u)$$ In particular $u_k\to u$ strongly in $BV(\hat{\Omega})$. Moreover, we see that $$\label{eq:Ef=Efk}
{\mathcal{E}}_{f_k}(u_k) \, =\, {\mathcal{E}}_f(u_k) \, \leq\ \, {\mathcal{E}}_f(u) + \a 2^{-k}\H^1(J_u).$$

*Step 2.* Let us approximate the function $u_k$. Let us fix $k\geq 0$ and ${\Omega}_i$. We can apply Lemma 4.1 of [@Bel_Cham_Gold] to the function $u_k{ \!\!\mbox{{ \Large$\llcorner$}} }{{\Omega}_i}$ and to the energy ${\mathcal{E}}_{f_k}(\cdot,{\Omega}_i)$. We obtain a sequence $(w^i_j)$ which enjoys the following properties: $$\label{eq:properties}
  \begin{aligned}
        & w^i_j({\Omega}_i)\subset u_k({\Omega}_i)\subset 2^{-k}{\mathbf{Z}},\quad \forall j\in{\mathbf{N}}, \mbox{ hence }w^i_j\in \P_k(\hat{\Omega}), \\
        & w^j_i\to u_k \mbox{ in } L^1({\Omega}_i)\mbox{ as }j{\rightarrow}+{\infty}, \\
        & 