Let be a sequence such that for all . Prove that is a Cauchy sequence.

How would I exactly go about this? My professor really didn't go over this. All she defined was that a Cauchy sequence is such that given there exists such that if , then .

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- Feb 17th 2010, 05:22 PMPinkkProve sequence is Cauchy
Let be a sequence such that for all . Prove that is a Cauchy sequence.

How would I exactly go about this? My professor really didn't go over this. All she defined was that a Cauchy sequence is such that given there exists such that if , then . - Feb 17th 2010, 05:37 PMmabruka
Given let be such that .

Then for every (supose without loss of generality that m<n) we have that

what do you think?

Errrr WAIT something is not right with the last inequality - Feb 17th 2010, 05:42 PMPinkk
I understand the inequality up until the point where you arrive at . How does the left side of the inequality lead to that, and how does that imply it is less than if we let ?

- Feb 17th 2010, 05:46 PMmabruka
Yeah i am sorry i got distracted there!(Happy) !

Let me edit the mistake. - Feb 17th 2010, 06:00 PMmabruka
About how we got to we aplied the hypothesis for each term. How many terms we had? n-m terms.

What i mean is that every term has the form

because m<m+a

About hte other issue, i am thinking maybe we need another bound there because i cant see how to continue form here. - Feb 17th 2010, 06:14 PMPinkk
Hmm I understand that there are terms, but I'm still not following how that inequality holds. For instance, if , we'd have

So we have that whole addition of those absolute values must be less than . But how is THAT less than ?

Edit: Nevermind, I see how each term must be less than or equal to which is added together times.

So I understand that, but yes, how does that lead to the inequality involving the epsilon? - Feb 17th 2010, 06:23 PMmabruka
Okey , first choose N such that . Then for ,

I think it is correct now :) - Feb 17th 2010, 06:33 PMPinkk
Thanks!

- Feb 17th 2010, 06:49 PMPinkk
One more quick question; would this be a Cauchy sequence if we replace with ?

- Feb 17th 2010, 08:30 PMDrexel28
- Feb 17th 2010, 08:36 PMPinkk
I don't think so, but I am not sure how to actually show it (at least only based on the knowledge we have learned so far in class). Because then we'd have (if we assume WLOG that , , and in order for that to be Cauchy, that whole inequality must also be less than any given where exists and . But if that were true for ALL , then would have to be less than some number for ALL n,m, which doesn't look like the case...ugh, I have a vague idea but I have no idea how to formulate into coherent/logical/proof-worthy words. (Headbang)(Headbang)