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I have a sequence $a_0, a_1, \ldots$ which has the following properties

$0 \leq a_i \leq 1$ for $i \geq 0$

$|a_{k+1} - a_k| < C^k$ for $k \geq 1$, where $C \in (0,1)$

Can I show that the sequence converges?

Thanks

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If $m\geq n$ then by triangle inequality we have $$|a_m-a_n|\leq \sum_{k=n}^{m-1}|a_{k+1}-a_k|\leq \sum_{k=n}^{m-1}C^k=C^n\frac{1-C^{m-n}}{1-C}\leq\frac{C^n}{1-C}\to_{n\to\infty}0$$ hence the $(a_k)$ is a Cauchy sequence and then it converges

Added Let $\epsilon>0$, since $\displaystyle \lim_{n\to\infty}\frac{C^n}{1-C}$ then there's $p\in\mathbb{N}$ and if $m\geq n\geq p$ we have $$|a_m-a_n|\leq \frac{C^n}{1-C}\leq\epsilon$$ and this explain why $(a_n)$ is a Cauchy sequence.

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  • $\begingroup$ How is it a Cauchy sequence? Can you explain? I know only the epsilon based definition of Cauchy sequence. $\endgroup$ – damned Apr 13 '13 at 6:57
  • $\begingroup$ @damned I added an explication why $(a_n)$ is a Cauchy sequence. $\endgroup$ – user63181 Apr 13 '13 at 13:52

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