Let $(\vec{x}_k)_{k=1}^\infty\subseteq \mathbb{R}^n$ be a sequence satisfying: $\exists$ constants $c\in (0, \infty)$ and $\gamma \in (0,1)$ such that $\|\vec{x}_k - \vec{x}_{k+1}\|\le c\gamma^k, \forall k\in\mathbb{N}$. Then $(\vec{x}_k)_{k=1}^\infty$ converges to some limit $\vec{a}\in \mathbb{R}^n$.
Proof:
It was proved earlier that a sequence which satisfies the condition above also satisfies the following:
$$\| \vec{x}_p-\vec{x}_q \|\le c\gamma^p/(1-\gamma),\forall 1\le p\le q\in \mathbb{N}$$ Now, $$\lim\limits_{p\to\infty\\q\to\infty} \|\vec{x}_p-\vec{x}_q\| \le \lim\limits_{p\to\infty\\q\to\infty} c\gamma^p/(1-\gamma)=0,$$ hence $(\vec{x}_k)_{k=1}^\infty$ is Cauchy, which implies that it is convergent.
I would appreciate your feedback on my proof approach. I've also tried to prove it via an $\varepsilon$ approach, but didn't find a way about it.