Limit set of subgroup of Hyperbolic Isometry

Question

When working with subgroup of Klenian groups (or in general just asking for discrete subgroup forgetting about the dimension of Hyperbolic space) the classic definition of limit set $\Lambda(\Gamma)$ is given as the set of accumulation point in the boundary of the orbit $\Gamma(x)$ for a chosen $x\in\mathbb{H}^n$. I have seen it stated in multiple reference but I have never seen it proven explicitly. Now the definition of such a set should not depend on the choice of the $x\in\mathbb{H}^n$. Is it correct to show that by a metric argument such as the following? Fix $x\neq y \in \mathbb{H}^{n},$ as point in Hyperbolic Space (a metric space) I have that $d_{\mathbb{H}^n}(x,y)=d<\infty$ for some real number $d$. Now I consider $\gamma_n(x)\to p\in\partial\mathbb{H}^n.$ Because I am acting via isometry I have that $\gamma_n(y)$ is always at distance $d$ from $\gamma_n(x)$, hence it is in any half-space of $p$ containing $\gamma_n(x).$ But this half-spaces are a system of neighborhoods for $p$, hence the points converge to the same point in the boundary. I hope my argument is, if not correct, at least clear. Thanks in advance for any hint or help.