\ell(B_n),\ell(C_n)=\mathcal{O}_{\omega}(d_n)\nonumber 
    \end{aligned}$$ Hence we get $|H_n|=\Theta_{\omega}(d_n)$. By $\color{blue}{\ref*{S}}$, we also have, $$\begin{aligned}
\label{C1} 
    f_G(|H_n|)=o_{\omega}(|H_n|)=o_{\omega}(d_n).
    \end{aligned}$$ Finally we get that $A$ is contained in the union of other two sides in $Con^{\omega}(X,e,d)$, as $\beta_n=o_{\omega}(d_n)$ (combining $\color{blue}{(\ref*{A1}),(\ref*{B1}),(\ref*{C1})}$). This contradicts out assumption that the triangle $\triangle_{\infty}$ is simple. Hence $Con^{\omega}(X,e,d)$ is an $\mathbb{R}$-tree. Which implies that the group $G$ is lacunary hyperbolic.

One can use similar proof to characterize lacunary hyperbolic spaces i.e, a geodesic metric space $(X,d)$ is lacunary hyperbolic if and only if the corresponding hyperbolicity function $f_X$ satisfies $\liminf_{t\rightarrow \infty}\frac{f_X(t)}{t}=0$.

Let $(X,d_X),(Y,d_Y)$ be metric spaces and $f:X\rightarrow Y$ be a map. The map $f$ is called a $(\lambda,c)$ quasi isometric embedding if for all $x,y\in X$,

$\frac{d_X(x,y)}{\lambda}-c < d_Y(f(x),f(y))< \lambda d_X(x,y)+c$

A $(\lambda,c)$ quasi isometric embedding is a $(\lambda,c)$ quasi-isometry if there exists a constant $D\geq0 $ such that for all $y\in Y$ there exists $x\in X$ with $d_Y(f(x),Y)<D$.

We say $X$ is quasi isometric to $Y$ is there exists a quasi isometry from $X$ to $Y$ for some $\lambda$ and $c$.

[@BH99] Quasi isometry is an equivalence relation.

\[quasigeodesic\] A $(\lambda,c)$quasi-geodesic in a metric space $(X,d_X)$ is a $(\lambda,c)$ quasi-isometric embedding of $\mathbb{R}$ into $(X,d_X)$. More precisely a map 