We can cover $\tau$ by polytopes $(\Delta_{m})_{m\in\mathbb{N}}$ such that $\Delta_{m-1}\subseteq\Delta_{m}$. By [BPS14, Proposition 2.5.24] for each $m$ there is a family of rational piecewise affine linear convex functions $(h_{i}^{m})_{i\in\mathbb{N}}$ on $\Delta_{m}$ converging uniformly to $h\Big|_{\Delta_{m}}$ (note that after normalization we can assume that $\mathbb{Z}$ is contained in the value group of $K$). We extend these functions to rational piecewise affine linear convex functions on $\overline{\tau}$. Then by Proposition 5.6 the metrics induced by the $h_{i}^{m}$ are semipositive piecewise $\mathbb{Q}$-linear metrics on $p_{\mathfrak{X}}^{-1}(\tau)$ which implies that the metric induced by $h\Big|_{\Delta_{m}}$ is semipositive. By Corollary 5.5 we have

$$c_{1}\left(\overline{\mathcal{O}}^{h_{i}^{m}\circ p_{\mathfrak{X}}}\right)^{n}=\Deg(S)\cdot n!\cdot\MA(h_{i}^{m})$$

for every $m,i\in\mathbb{N}$. Denoting the interior of $\Delta_{m}$ by $\Delta_{m}^{\circ}$ and using Proposition 4.13 we find that for fixed $m$ the left hand side converges to $c_{1}\left(\overline{\mathcal{O}}^{h\circ p_{\mathfrak{X}}}\right)^{n}$ on $p_{\mathfrak{X}}^{-1}(\Delta_{m}^{\circ})$. The right hand side converges to $\Deg(S)\cdot n!\cdot\MA(h)$ on $\Delta_{m}^{\circ}$ by continuity of the real Monge-Ampère operator. As this holds for any $m$ and the $\Delta_{m}$ cover $\tau$ this proves the corollary. ∎