By Vojta’s version of Nagata’s compactification theorem ([Voj, Theorem 5.7]) we may assume that $X$ is proper. We show by reverse induction over $m\in\{0,...,n\}$ that the claim holds when for some choice of pairwise different $i_{1},...,i_{n}\in\{1,...,n\}$ the sequences $\Big(\|\cdot\|_{i_{1},k}\Big)_{k\in\mathbb{N}},...,\Big(\|\cdot\|_{i_{m},k}\Big)_{k\in\mathbb{N}}$ are constant with respect to $k$. The case $m=n$ is clear. So let $0\leq m<n$ and assume that the claim holds for $m+1$. For $j\in\{m+1,...,n\}$ we can write $\|\cdot\|_{i_{j},k}=\|\cdot\|_{i_{j},1}\otimes\|\cdot\|^{\prime}_{j,k}$ for a sequence of piecewise $\mathbb{Q}$-linear metrics $\Big(\|\cdot\|^{\prime}_{j,k}\Big)_{k\in\mathbb{N}}$ on $\mathcal{O}_{X^{\textup{an}}}\Big|_{V}$ converging uniformly to a continuous metric $\|\cdot\|^{\prime}_{j}$ on $\mathcal{O}_{X^{\textup{an}}}\Big|_{V}$. Denote by $\overline{\mathcal{O}}_{j,k}$ the line bundle $\mathcal{O}_{X^{\textup{an}}}\Big|_{V}$ endowed with the metric $\|\cdot\|^{\prime}_{j,k}$. We show that

$$\left(\mu_{m,k}:=c_{1}(\overline{L}_{i_{1},1})\wedge...\wedge c_{1}(\overline{L}_{i_{m},1})\wedge c_{1}(\overline{L}_{i_{m+1},k})\wedge...\wedge c_{1}(\overline{L}_{i_{n},k})\right)_{k\in\mathbb{N}}$$

is a Cauchy sequence with respect to the weak topology on the space of Borel-measures on $V$. Thus we have to show that for all continuous functions $f$ on $X$ with compact support in $V$:

$$\left|\int_{V}f\;\mu_{m,k}-\int_{V}f\;\mu_{m,k^{\prime}}\right|\underset{k,k^{\prime}\rightarrow\infty}{\longrightarrow}0.$$

Let $W$ be a compact strictly $K$-analytic domain with $\supp(f)\subseteq\overset{\circ}{W}$ and $W\subseteq V$. By [GM19, Proposition 2.7] we may extend the metrics from $W$ to $X^{\textup{an}}$ and hence assume that they are defined on the whole space. Hence by Chow’s lemma and the projection formula we may assume that $X$ is projective. Then by [Gub03, Proposition 10.5] any formal model of $X$ is dominated by a projective model. Any formal line bundle on this model becomes semipositive after tensoring with $\mathcal{O}(n)$ for $n$ big enough by using Serre’s theorem ([Har77, Theorem II.5.17]) on the special fibre. As a consequence one can write any formal metric on any line bundle on $X$ as a quotient of two semipositive formal metrics (on possibly different line bundles). We will see below, that $\mu_{m,k}(Z)$ is bounded with respect to $k$ for every compact subset $Z\subseteq V$. Hence, as the set of piecewise $\mathbb{Q}$-linear metrics is dense in the space of continuous metrics on $\mathcal{O}_{X^{\textup{an}}}$ with respect to uniform convergence (Proposition 3.13), we may assume that $f=-\log\|1\|$ for a formal metric $\|\cdot\|$ on $\mathcal{O}_{X^{\textup{an}}}$. Then we can write $\|\cdot\|=\|\cdot\|_{+}/\|\cdot\|_{-}$ for two semipositive formal metrics $\|\cdot\|_{+},\|\cdot\|_{-}$ on some line bundles $L_{+}$ respectively $L_{-}$ on $X^{\textup{an}}$. In fact $L_{+}=L_{-}$ but we will use the notation $\overline{L}_{+}$ and $\overline{L}_{-}$ to distinguish between the two metrics. Write $\overline{\mathcal{O}}_{X}^{f}$ for the line bundle $\mathcal{O}_{X^{\textup{an}}}\Big|_{V}$ endowed with the metric $\|1\|=e^{-f}$ and to shorten notation $\mu_{m}:=c_{1}(\overline{L}_{i_{1},1})\wedge...\wedge c_{1}(\overline{L}_{i_{m},1})$ which is a purely formal notation. Furthermore without loss of generality assume $i_{1}=1,...,i_{m}=m$. We have

	$\displaystyle\Big|$	$\displaystyle\int_{V}f\;\mu_{m,k}-\int_{V}f\;\mu_{m,k^{\prime}}\Big|$
		$\displaystyle=\Big|\sum_{i=1}^{n-m}\int_{V}f\;\mu_{m}\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{m+i,k}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)$
		$\displaystyle\phantom{\Big|\sum_{i=1}^{n}}-\int_{V}f\;\mu_{m}\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{L}_{m+i,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)\Big|$
		$\displaystyle=\Big|\sum_{i=1}^{n-m}\int_{V}f\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{L}_{m+i,1}\otimes\overline{\mathcal{O}}_{m+i,k}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...$
		$\displaystyle\phantom{\Big|\sum_{i=1}^{n}}-\int_{V}f\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{L}_{m+i,1}\otimes\overline{\mathcal{O}}_{m+i,k^{\prime}}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...\Big|$
		$\displaystyle=\Big|\sum_{i=1}^{n-m}\int_{V}f\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{m+i,k}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...$
		$\displaystyle\phantom{\Big|\sum_{i=1}^{n}}-\int_{V}f\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{m+i,k^{\prime}}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...\Big|$

Since the support of $f$ is contained in $V$ and by Lemma 4.8 these last integrals depend only on the restrictions of the metrics to $V$. Hence we may instead consider them as integrals over $X^{\textup{an}}$ which allows us to use Lemma 4.6 as $X^{\textup{an}}$ has no boundary ([Ber90, Theorem 3.4.1]). In combination with an index shift, the last term amounts to

	$\displaystyle\Big|$	$\displaystyle\sum_{i=m+1}^{n}\int_{X^{\textup{an}}}-\log\|1\|^{\prime}_{i,k}\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...$
		$\displaystyle-\int_{X^{\textup{an}}}-\log\|1\|^{\prime}_{i,k^{\prime}}\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\Big|$

As any point in $X^{\textup{an}}\setminus\supp(f)$ has a strictly $K$-analytic neighbourhood on which $f$ vanishes, the support of $\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...$ is contained in $\supp(f)$ by Lemma 4.8. So the last display equals

	$\displaystyle\Big|$	$\displaystyle\sum_{i=m+1}^{n}\int_{\supp(f)}-\log\|1\|^{\prime}_{i,k}\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...$
		$\displaystyle-\int_{\supp(f)}-\log\|1\|^{\prime}_{i,k^{\prime}}\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\Big|$
		$\displaystyle=\Big|\sum_{i=m+1}^{n}\int_{\supp(f)}\log\Big(\|1\|^{\prime}_{i,k^{\prime}}/\|1\|^{\prime}_{i,k}\Big)\;...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\Big|$
		$\displaystyle\leq 2\cdot\sum_{i=m+1}^{n}\sup_{x\in\supp(f)}\left|\log\Big(\|1\|^{\prime}_{i,k}(x)/\|1\|^{\prime}_{i,k^{\prime}}(x)\Big)\right|$
		$\displaystyle\phantom{\leq\sum_{i=m+}^{n}}\cdot\max_{s\in\{+,-\}}\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{L}_{s}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...(\supp(f))\underset{k,k^{\prime}\rightarrow\infty}{\longrightarrow}0.$

Here the last term converges to zero as $\sup_{x\in\supp(f)}\left|\log\Big(\|1\|^{\prime}_{i,k}(x)/\|1\|^{\prime}_{i,k^{\prime}}(x)\Big)\right|$ tends to zero by uniform convergence of $\|\cdot\|^{\prime}_{i,k}$ and compactness of $\supp(f)$ and $\mu_{m}\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{L}_{s}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)$ are positive measures on $V$ which converge by the induction hypothesis weakly to a positive Radon measure which implies that their mass of $\supp(f)$ is bounded with respect to $k,k^{\prime}$. To go into more detail, let $g$ be a continuous non-negative function on $V$ with compact support such that $g(x)>1$ for all $x\in\supp(f)$. The existence of such a function follows for example from a partition of unity argument ([Flo03, 1.5.1]) applied to the open cover $\{\overline{V}\setminus\supp(f),V\}$ of the closure $\overline{V}$ of $V$ (note that $\overline{V}$ is compact as $X$ is proper over $K$). Then

	$\displaystyle\mu_{m}$	$\displaystyle\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{L}_{s}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)(\supp(f))$
		$\displaystyle\leq\int g\;\mu_{m}\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{L}_{s}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)$

where the last term converges for $k,k^{\prime}\rightarrow\infty$ and is hence bounded with respect to $k,k^{\prime}$.
We now define a positive linear functional on the space of continuous functions with compact support in $V$ by

	$\displaystyle C_{c}(V)$	$\displaystyle\rightarrow\mathbb{R}_{\geq 0},$
	$\displaystyle f$	$\displaystyle\mapsto\lim_{k\rightarrow\infty}\int_{V}f\;\mu_{m,k}.$

By the Riesz Representation Theorem ([Rud87, Theorem 2.14]) this corresponds to a positive Radon measure $\mu$ on $V$ and we have $\mu_{m,k}\rightarrow\mu$ weakly for $k\rightarrow\infty$.
It remains to show that $\mu_{m,k}(Z)$ is bounded with respect to $k$ for every compact subset $Z\subseteq V$. So let $Z\subseteq V$ be compact and $f$ a continuous non-negative function on $V$ with compact support such that $f(x)>1$ for all $x\in Z$. As above the existence of such a function follows from a partition of unity argument ([Flo03, 1.5.1]) applied to the open cover $\{\overline{V}\setminus Z,V\}$ of the closure $\overline{V}$ of $V$. Again we may assume that $f$ is a model function, i.e. of the from $-\log\|\cdot\|$ for a piecewise $\mathbb{Q}$-linear metric $\|\cdot\|$ on $\mathcal{O}_{X^{\textup{an}}}$ (we can even assume that $\|\cdot\|$ is a formal metric) and we use the same notation as above. To be more precise, let $\epsilon>0$ such that $f(x)>1+\epsilon$ for all $x\in Z$. First extend $f$ to $X^{\textup{an}}$ by zero and then define a new function $\tilde{f}$ by $\tilde{f}(x)=f(x)-\epsilon/2$. By Proposition 3.13 we may approximate $\tilde{f}$ by a model function $\phi$ such that $|\phi(x)-\tilde{f}(x)|<\epsilon/2$ for all $x\in X^{\textup{an}}$. Then by [GM19, Proposition 2.12 (d)], $\max\{0,\phi\}$ is a model function on $X^{\textup{an}}$ with compact support in $V$ which is greater than one at $Z$. We have

	$\displaystyle\sup_{k\in\mathbb{N}}\mu_{m,k}(Z)$	$\displaystyle\leq\sup_{k\in\mathbb{N}}\int_{V}f\;\mu_{m,k}$
		$\displaystyle=\sup_{k\in\mathbb{N}}\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{L}_{m+1,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle=\sup_{k\in\mathbb{N}}\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{L}_{m+1,1}\otimes\overline{\mathcal{O}}_{m+1,k})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle=\sup_{k\in\mathbb{N}}\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{L}_{m+1,1})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle\phantom{\lim_{k\rightarrow\infty}}+\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{\mathcal{O}}_{m+1,k})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$

Again using Lemma 4.6 and the same argumentation as above for the second summand this amounts to

	$\displaystyle\sup_{k\in\mathbb{N}}$	$\displaystyle\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{L}_{m+1,1})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle\phantom{\lim_{k\rightarrow\infty}}+\int_{\supp(f)}-\log\|1\|^{\prime}_{m+1,k}\;\mu_{m}\wedge c_{1}(\overline{\mathcal{O}}_{X}^{f})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle\leq\sup_{k\in\mathbb{N}}\sup_{x\in V}f(x)\cdot\mu_{m}\wedge c_{1}(\overline{L}_{m+1,1})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})(\supp(f))$
		$\displaystyle\phantom{\lim_{k\rightarrow\infty}}+\sup_{k\in\mathbb{N}}\sup_{x\in\supp(f)}\left|\log\Big(\|1\|^{\prime}_{m+1,k}(x)\Big)\right|$
		$\displaystyle\phantom{\lim_{k\rightarrow\infty}+}\cdot 2\cdot\max_{s\in\{+,-\}}\mu_{m}\wedge c_{1}(\overline{L}_{s})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})(\supp(f))$

By the induction hypothesis all measures appearing in this last term converge for $k\rightarrow\infty$. Hence the measure of $\supp(f)$ is bounded with respect to $k$. Furthermore $\sup_{x\in V}f(x)<\infty$ as $f$ has compact support in $V$ and $\sup_{x\in\supp(f)}\left|\log\Big(\|1\|^{\prime}_{m+1,k}(x)\Big)\right|$ is bounded with respect to $k$ by uniform convergence of $\Big(\|\cdot\|^{\prime}_{m+1,k}\Big)_{k\in\mathbb{N}}$ and compactness of $\supp(f)$. We conclude that the last term is bounded with respect to $k$. This proves the induction step. The claim is then the case $m=0$. ∎