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Let $\{an\}$ be a sequence of real numbers with the property that there exists a constant $C, 0 < C < 1$, such that $|a_{n+2} − a_{n+1}| ≤ C|a_{n+1} − a_n|$. Show that $\{an\}$ is a Cauchy sequence.

I think it's clear from common sense, since as $0 < C < 1$ then the terms get gradually closer and closer since this is a recursive sequence, and this is essentially what the definition of a Cauchy sequence. Thanks for any help.

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math.stackexchange.com/questions/668736/… –  user61527 May 26 '14 at 15:02
This sounds like a job for.... The triangle inequality! –  mixedmath May 26 '14 at 15:07

1 Answer 1

This is a very classic problem. By the hypothesis we have by induction:

$$|a_{n+1}-a_n|\le C^n|a_1-a_0|$$ so we have by the triangle inequality $$|a_{n+p}-a_n|\le\sum_{k=1}^p|a_{n+k}-a_{n+k-1}|\\\le|a_1-a_0|C^n\sum_{k=1}^pC^{k-1}=|a_1-a_0|C^n\frac{1-C^p}{1-C}\le|a_1-a_0|\frac{C^n}{1-C}\xrightarrow{n\to\infty}0$$ and now we can conclude easily.

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Thanks for your answer but in the first line, did you mean to write $C^n|a_{n+1} - a_{n}|$? –  Crockett May 26 '14 at 15:17
No I mean what I wrote. Try to prove it by induction! –  Sami Ben Romdhane May 26 '14 at 15:19

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