Lemma 17.10.90.6C6200F67BD7 For any $\ell ,\ell '\in \mathcal{R}^{\infty }(X,\theta ;\phi )$, we have
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17.10.91
\begin{equation} d_1(\ell ,\ell ')\leq d_1(\ell ,\ell \lor \ell ')+d_1(\ell ',\ell \lor \ell ')\leq C_ nd_1(\ell ,\ell '), \end{equation}
where $C_ n=3(n+1)2^{n+2}$.
Proof.
The first inequality is trivial. As for the second, we estimate
\[ \begin{aligned} d_1(\ell ,\ell \lor \ell ')=& \mathbf{E}^{\phi }(\ell \lor \ell ')-\mathbf{E}^{\phi }(\ell )& \textup{by proposition \href{../../tag/3A33DCAAE061/}{17.10.83}}\\ =& \lim _{t\to \infty } \frac{1}{t}E_{\theta }^{\phi }\left(\ell _ t\lor \ell '_ t\right)-\mathbf{E}^{\phi }(\ell )& \textup{by (\href{../../tag/CA6EEFF029A5/}{17.10.87})}\\ =& \lim _{t\to \infty } \frac{1}{t} d_1\left(\ell _ t\lor \ell '_ t,\ell _ t\right)& \textup{by (\href{../../tag/D8C8C0BB6585/}{17.10.42})}. \end{aligned} \]
By symmetry, we find
\[ d_1(\ell ,\ell \lor \ell ')+d_1(\ell ',\ell \lor \ell ')\leq \lim _{t\to \infty } \frac{1}{t} \Bigl (d_1\left(\ell _ t\lor \ell '_ t,\ell _ t\right)+d_1\left(\ell _ t\lor \ell '_ t,\ell '_ t\right)\Bigr ). \]
By theorem 17.10.48 and 17.10.47, for each $t>0$,
\[ d_1\left(\ell _ t\lor \ell '_ t,\ell _ t\right)+d_1\left(\ell _ t\lor \ell '_ t,\ell '_ t\right)\leq 3(n+1)2^{n+2} d_1(\ell _ t,\ell '_ t). \]
Now (17.10.91) follows.
$\square$
Pome
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