Let $x_n$ be a real sequence.
Suppose that there is an $a>1$ such that $|x_{n+1} -x_n|\le a^{-n}$ for all $n\in\Bbb N$. Prove that $x_n \to x$ for some $x \in\Bbb R$.
Let $x_n$ be a real sequence.
Suppose that there is an $a>1$ such that $|x_{n+1} -x_n|\le a^{-n}$ for all $n\in\Bbb N$. Prove that $x_n \to x$ for some $x \in\Bbb R$.