Let $u$ and $v$ be two points of $\mathcal{H}$. The hyperbolic distance between $u$ and $v$ is defined by $$d(u,v) = \inf_\gamma \ell(\gamma)$$ where the infimum is taken over the set of continuous and piecewise $\mathcal{C}^1$ paths $\gamma:[a,b]\rightarrow\mathcal{H}$ such that $\gamma(a)=u$ and $\gamma(b)=v$. Show that $d$ is a distance on $\mathcal{H}$, that is,
$d(u,v) = d(v,u)$,
$d(u,w) \leq d(u,v)+d(v,w)$, and
$d(u,v)=0 \Leftrightarrow u=v$
for all $u,v,w\in\mathcal{H}$.
Let $u$ and $v$ be two points of $\mathcal{H}$. The hyperbolic distance between $u$ and $v$ is defined by
$$d(u,v) = \inf_\gamma \ell(\gamma)$$
where the infimum is taken over the set of continuous and piecewise $\mathcal{C}^1$ paths $\gamma:[a,b]\rightarrow\mathcal{H}$ such that $\gamma(a)=u$ and $\gamma(b)=v$.
Show that $d$ is a distance on $\mathcal{H}$, that is,
\begin{itemize}
\item $d(u,v) = d(v,u)$,
\item $d(u,w) \leq d(u,v)+d(v,w)$, and
\item $d(u,v)=0 \Leftrightarrow u=v$
\end{itemize}
for all $u,v,w\in\mathcal{H}$.