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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 $x \in\Bbb R$.

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2 Answers 2

HINT: Show that the hypothesis implies that the sequence is Cauchy. A basic knowledge of geometric series is useful.

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Since $a>1$, $\frac{1}{a} < 1$ and $\frac{1}{a^n} > \frac{1}{a^{n+1}}$. Then, for any $\epsilon > 0$ exists $n$ that makes your sequence a Cauchy sequence.

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