ScalingStacks

Proof. For any $x\in X^{\mathrm{an}}$ and any $e_{1}(x)\in V_{1}(L)(x)$, we have

$$\begin{split}\vvvert e_{1}(x)\vvvert_{(X|x);\mathrm{sp}}&=\lim_{\begin{subarray}{c}n\to\infty\end{subarray}}\vvvert e_{1}^{\otimes n}(x)\vvvert_{X|x}^{\frac{1}{n}}\\ &=\lim_{\begin{subarray}{c}n\to\infty\end{subarray}}\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})(e_{1}(x))(x)=\mathcal{P}(\vvvert\mathord{\cdot}\vvvert)(e_{1}(x))(x).\end{split}$$

∎