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Proposition 19.6.147B4552909292

Proposition 19.6.14.7B4552909292 Let $(\Delta ^ i)_{i\in I}$ be a decreasing net in $\mathcal{K}_ n$ such that $\Delta \subseteq \bigcap _{i\in I}\Delta ^ i$. Consider a decreasing net $(\Delta ^ i_{\bullet })_{i\in I}$ with $\Delta ^ i_{\bullet }\in \mathrm{TC}(\Delta ^ i)$ for all $i\in I$ such that there is $\Delta _{\bullet }\in \mathrm{TC}(\Delta )$ satisfying the following properties:

  1. A929D98DE380

    $\Delta _{\max }= \lim _{i\in I}\Delta ^ i_{\max }$;

  2. 791125E2CC9D

    for any $\tau <\Delta _{\max }$, we have $\Delta ^ i_{\tau }\xrightarrow {d_{\mathrm{Haus}}}\Delta _{\tau }$.

Then for any $a\in \Delta $, we have

E0766167D753 19.6.15
\begin{equation} \lim _{i\in I} G[\Delta ^ i_{\bullet }](a)=G[\Delta _{\bullet }](a). \end{equation}

Proof. Fix $a\in \Delta $. It follows immediately from the definition of $G$ that the net $(G[\Delta ^ i_{\bullet }](a))_{i\in I}$ is decreasing and the $\geq $ direction in (19.6.15) holds. Let us prove the reverse inequality. Let $\tau $ denote the left-hand side of (19.6.15) for the moment. By definition, for any $\epsilon >0$ and any $i\in I$, we have $a\in \Delta ^ i_{\tau -\epsilon }$. It follows that

\[ a\in \Delta _{\tau -\epsilon }. \]

Therefore,

\[ \tau \leq G[\Delta _{\bullet }](a). \]

This completes the proof. $\square$

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