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We're supposed to prove or disprove the statement that $(1 + (-1)^n)$ is a Cauchy sequence. I don't think it is because this function oscillates and doesn't converge, so I don't think it's Cauchy. Why or why not is this a Cauchy sequence? Thanks.

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Hint: Cauchy $\iff$ convergent, for sequences of real numbers –  Jean-Claude Arbaut Dec 7 '13 at 0:32
    
How could I show that this sequence is not convergent? –  anonymous Dec 7 '13 at 0:35
    
It has two subsequences that converge to different values (actually they are constant). Try with terms with odd and even indices. –  Jean-Claude Arbaut Dec 7 '13 at 0:36

3 Answers 3

Note that $\left \vert a_{n+1} - a_n \right \vert = 2$ for all $n$. So is this sequence Cauchy?

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You can apply the definition of Cauchy sequence directly. Note that if $a_n=1+(-1)^n$, then

$$a_n=\begin{cases} 2,&\text{if }n\text{ is even}\\ 0,&\text{if }n\text{ is odd}\;. \end{cases}$$

Let $\epsilon>0$. If the sequence is Cauchy, there must be an $m\in\Bbb N$ such that $|a_k-a_\ell|<\epsilon$ whenever $k,\ell\ge m$. Now consider $|a_k-a_{k+1}|$ for any $k\in\Bbb N$; it’s equal to ... ?

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If the series converges to a limit $L$, then for sufficiently big values of $n$, you'd have $(1+(-1))^n$ between $L\pm1/10$. Thus two consecutive terms would differ from each other by less than $2/10$. But $0$ and $2$ do not differ by less than $2/10$. Hence the sequence diverges. Therefore it's not a Cauchy sequence.

PS: There are some who say that although some sequences are Cauchy sequences, no sequence is Cauchy, since Cauchy is a man who lived in the 19th century. The idea is that the term "Cauchy sequence" is a compound noun, rather than "Cauchy" being an adjective. I sympathize with this view, but mathematicians generally don't even suspect there is such a view. Language is messy.

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I suppose the height of reputation comes when your surname becomes an adjective, like Freudian or Kafkaesque. –  anonymous Dec 7 '13 at 1:13
    
"Freudian" and "Kafkaeque" are adjectives because of their suffixes; "Freud" and "Kafka" are not. –  Michael Hardy Dec 7 '13 at 1:37
    
"no sequence is Cauchy, since Cauchy is a man who lived in the 19th century." That's a strange argument: there was and is more than one person named "Cauchy". "Cauchy sequence" is a mathematical term: it gets a formal definition. The standard one of course is "A sequence is Cauchy if..." According to this definition, "Cauchy" is clearly an adjective. –  Pete L. Clark Dec 7 '13 at 1:39
    
You don't have to like the terminology or even use it: an internet search shows that a few people speak of "Cauchian sequences", so you could try that. But there is absolutely nothing illicit about the standard use of the term: a word is an adjective if it is defined and used that way, period. This usage is not unique to mathematics: e.g. people say "Allen wrench" and no one says "hand me an Allen", so "Allen" is an adjective here. –  Pete L. Clark Dec 7 '13 at 1:41
    
@PeteL.Clark : A good definition would say is "A Cauchy sequence is....". Yes, it is being used as an adjective if you insist on saying "a sequence is Cauchy". But read my whole comment. –  Michael Hardy Dec 7 '13 at 1:41

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