ScalingStacks

Proof. Recall that one can find $M\in\mathbb{N}$ such that $L^{\otimes M}$ is very ample and for any $n\geq M$, the restriction map from $V_{n}(L)$ to $V_{n}(L|_{Y})$ is surjective.

By the asymptotic Fubini-Study assumption, there exist norms $\{\lVert\mathord{\cdot}\rVert_{n}\}_{n\in\mathbb{N}}$ on $V_{n}(L)$ such that $\phi$ is given by $\mathcal{P}(\{\lVert\mathord{\cdot}\rVert_{n}\}_{n\in\mathbb{N}})$. Let $\psi_{n}$ denote the metric $\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert)_{n}$ on $L$. By the continuity assumption, the convergence to envelop metric is uniform (see §3.1 12.). So for any $\epsilon>0$, there exists $M^{\prime}\geq M$ such that

$$\dist(\psi_{M^{\prime}},\phi)\leq\frac{1}{3}\epsilon,$$

hence for any $n\in\mathbb{N}$, one has

$$\dist(\lVert\mathord{\cdot}\rVert_{n\psi_{M^{\prime}}},\lVert\mathord{\cdot}\rVert_{n\phi})\leq\frac{1}{3}n\epsilon,\quad\dist(\lVert\mathord{\cdot}\rVert_{n\psi_{M^{\prime}},X|Y},\lVert\mathord{\cdot}\rVert_{n\phi,X|Y})\leq\frac{1}{3}n\epsilon.$$

By Corollary 5.8, the Banach algebra $\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},(\psi_{M^{\prime}})_{X|Y})$ is an affinoid algebra. By Proposition 5.9, there exists $C_{M^{\prime}}(X,Y,\phi)>0$ such that

$$\forall\underline{t}\in V_{{\scriptscriptstyle\bullet}}(L_{X|Y}),\quad\vvvert\underline{t}\vvvert_{\psi_{M^{\prime}},X|Y}\leq C_{M^{\prime}}\cdot\vvvert\underline{t}\vvvert_{\psi_{M^{\prime}},X|Y;\mathrm{sp}}=\vvvert\underline{t}\vvvert_{\psi_{M^{\prime}},Y}.$$

Combining this comparison with above estimates, one has that for every $n\geq n_{Y}:=\lceil\ln(C_{M^{\prime}})/(\epsilon/3)\rceil$

$$\begin{split}\lVert t_{n}\rVert_{n\phi,X|Y}&\leq\mathrm{e}^{\frac{1}{3}n\epsilon}\cdot\lVert t_{n}\rVert_{n\psi_{M^{\prime}},X|Y}\\ &\leq\mathrm{e}^{\frac{1}{3}n\epsilon}\cdot C_{M^{\prime}}\cdot\lVert t_{n}\rVert_{n\psi_{M^{\prime}}|_{Y}}\\ &\leq\mathrm{e}^{\frac{2}{3}n\epsilon}\cdot C_{M^{\prime}}\cdot\lVert t_{n}\rVert_{n\phi|_{Y}}\\ &\leq\mathrm{e}^{n\epsilon}\cdot\lVert t_{n}\rVert_{n\phi|_{Y}}.\end{split}$$

Hence there exists $s_{n}\in V_{n}(L)$ with $\lVert s_{n}\rVert_{n\phi}\leq\mathrm{e}^{n\epsilon}\cdot\lVert t_{n}\rVert_{n\phi|_{Y}}$. ∎