ScalingStacks

Proof. Choose an arbitrary $\boldsymbol{\delta}$ with $|\boldsymbol{\delta}|\leq\epsilon$. Then

$$\dist(\lVert\mathord{\cdot}\rVert_{\phi},\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})})\leq\epsilon.$$

By Proposition 3.12, we have

$$\dist(\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi}),\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})}))=\dist(\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi}),\phi(\boldsymbol{\delta}))\leq\epsilon,$$

by the assumption and Proposition 3.11,

$$\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi})=\phi,$$

so the conlusion holds. ∎