Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $x \in\Bbb R$.

share|improve this question
    
What have you tried? –  icurays1 Nov 30 '12 at 23:22
add comment

2 Answers

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

share|improve this answer
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.