Proof. [00M3]

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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. ∎