from different definition. For proofs we refer [@BH99], chapter $III$.\
Next property of hyperbolic space can be deduce from cutting hyperbolic $n$-gon into triangles.

[@Ol93]\[hyperbolicngon\] For $n\geq 3$, any side of a geodesic $n$-gon in a $\delta$-hyperbolic space belongs to the $(1+\log_2(n-1))\delta$ neighborhood of the other $(n-1)$ sides.

Ultrafilter and asymptotic cone
-------------------------------

In a metric space a segment is defined to be a subset isometric to an interval of the set of real numbers $\mathbb{R}$. A ray is defined to be a segment isometric to $[0,\infty)$.

An $\mathbb{R}$-tree is a complete metric space $T$ such that, for all points $x,y$ in $T$, there is a unique compact arc with endpoints $x$ and $y$, such that this arc is a segment.

A non principle ultrafilter $\omega$ is a finitely additive measure on all subsets $\mathcal{S}$ of natural numbers $\mathbb{N}$ such that , $\omega(\mathcal{S})\in \{0,1 \}, \omega(\mathbb{N})=1$ and $\omega(\mathcal{S})=0$ for all finite subset of $\mathbb{N}$.

For a bounded sequence of numbers $\{x_n \}_{n\in \mathbb{N}}$ the ultralimit $\lim^{\omega}_i$ with respect to $\omega$ is the unique number $a$ such that ; $$\omega(\{i\in \mathbb{N}| \ |a-x_i|<\epsilon \})=1, \ \ for \ all \ \epsilon>0$$ Similarly $\lim^{\omega}_i x_i=\infty$ if; $$\omega(\{i\in \mathbb{N}| \ x_i>M \})=1, \ 