Theorem 17.10.81B30037E401C6

Proof. Symmetry and the triangle inequality follow from the corresponding pointwise properties of $d_1$ on $\mathcal{E}^1(X,\theta ;\phi )$.

We next show that $d_1(\ell ,\ell ')<\infty $ for any $\ell ,\ell '\in \mathcal{R}^1(X,\theta ;\phi )$. Let $C_\ell $ be the constant in proposition 17.10.39, and set $\tilde{\ell }_ t=\ell _ t-C_\ell t$. Then $\tilde{\ell }$ is again a geodesic ray, $\sup _ X\tilde{\ell }_ t=0$, and hence $\tilde{\ell }_ t\leq \phi $ by proposition 16.3.21. Thus, by proposition 17.10.65,

Moreover, by (16.5.13), $d_1(\ell _ t,\tilde{\ell }_ t)=|C_\ell |t\int _ X\theta _{\phi }^ n$. Hence $d_1(\ell _ t,\phi )$ grows at most linearly in $t$. The same argument applies to $\ell '$, and the triangle inequality gives $d_1(\ell ,\ell ')<\infty $.

It remains to prove separation: suppose that $\ell ,\ell '\in \mathcal{R}^1(X,\theta ;\phi )$ and $d_1(\ell ,\ell ')=0$.

Fix $s>0$. Then it follows from proposition 17.10.79 that

Therefore, $\ell _ s=\ell '_ s$ and hence $\ell =\ell '$. $\square$