$$\begin{aligned}
\label{lac}
    f_G(d_n)=o(d_n).
    \end{aligned}$$ Let $\omega$ be an arbitrary non principle ultrafilter. According to lemma $\color{blue}{\ref*{limitgeodesictrg}}$ applied to the collection $\mathcal{P}$ of all one-element subsets of $Con^{\omega}(G,d)$ to show that $Con^{\omega}(G,d)$ is an $\mathbb{R}$-tree it is sufficient to prove that it contains no non-trivial simple limit geodesic triangles.\
Let $\triangle_{\infty}$ be a simple non-trivial limit geodesic triangle in $Con^{\omega}(G,d)$ with sides A, B and C with,

$\lim^{\omega}{A_n}=A, 
        \lim^{\omega}{B_n}=B, 
        \lim^{\omega}{C_n}=C$

where $A_n,B_n,C_n$ are geodesics in $X:=\Gamma(G,S)$ with endpoints $a_n',b_n$ for $A_n$ and $b_n',c_n$ for $B_n$ and $c_n',a_n$ for $C_n$, for all $n\in \mathbb{N}$. Let ${\triangle}_{\infty}$ is approached by hexagons $H_n$, formed by vertices $a_n{a_n}'b_n{b_n}'c_n{c_n}'$ in $X$ with $\ell (P^i_n)$[^3]$=o_{\omega}(d_n)$ for $i=1,2,3$, where, $ P^1_n$ is a geodesic joining $a_n,a_n'$ in $X$ and respectively $P^2_n$ is a geodesic joining $b_n,b_n'$ in $X$ and $P^3_n$ is a geodesic joining $c_n,c_n'$ in $X$.\
Denote the perimeter of the hexagon $H_n$ by $|H_n|$. Let $A$ be a nontrivial side of $\triangle_{\infty}$. By lemma $\color{blue}{\ref*{hyperbolicgeodesicngon1}}$, $A_n$ belongs to closed $4f_G(|H_n|)$ neighborhood of other $5$ sides in $H_n$. In particular $A_n$ is contained in closed $\beta_n$ neighborhood of $B_n\cup C_n $, where $$\begin{aligned}
\label{A1} 
    \beta_n=4f_G(|H_n|) + \max_{i=1,2,3}\{\ell(P^i_n)  \}.
    \end{aligned}$$ We have, $$\begin{aligned}
\label{B1}
    \ell(P^i_n)=o_{\omega}(d_n)\\
    \ell(A_n) = \Theta_{\omega}(d_n)\nonumber\\
    