Let us consider the smooth metric on $\mathscr{O}(1)$ associated to the model $(\mathfrak{X},\mathscr{O}(1),1)$ of $(\mathrm{P}^{n}_{K},\mathscr{O}(1))$. Let $\mathfrak{U}_{i}$ be the formal open subset of $\mathrm{P}^{n}_{K}$ defined by the non-vanishing of the homogeneous coordinate $x_{i}$. Over, $\mathfrak{U}_{i}$, $\mathscr{O}(1)$ has a global non-vanishing section, namely the one associated to the homogeneous polynomial $X_{i}$. The generic fiber $U_{i}$ of $\mathfrak{U}_{i}$ in the sense of algebraic geometry is an affine space, with coordinates $z_{j}=x_{j}/x_{i}$, for $0\leq j\leq n$, and $j\neq i$. However, its generic fiber $\mathrm{U}_{i}$ in the sense of rigid geometry is the $n$-dimensional polydisk in this affine space defined by the inequalities $\left|{z_{j}}\right|\leq 1$. We thus observe that for any $x\in(\mathfrak{U}_{i})_{K}$,

$$\left\|{X_{i}}\right\|(x)=1=\frac{1}{\max(\left|{z_{0}}\right|,\dots,\left|{z_{i-1}}\right|,1,\left|{z_{i+1}}\right|,\dots,\left|{z_{n}}\right|)}=\frac{\left|{x_{i}}\right|}{\max(\left|{x_{0}}\right|,\dots,\left|{x_{i}}\right|)}=\left\|{X_{i}}\right\|_{\mathrm{W}}(x).$$

In other words, the Weil metric on $\mathscr{O}(1)$ is a smooth metric.