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# cauchy sequences. watch

1. having a bit of bother proving the following:

For each positive integer n, let a_n be a real number with a_n[-11,23] and define the sequenceb by

show the sequence is cauchy.

only a 4 mark question on a paper of ~100 marks so it cant be that intensive a question, I think I'm just missing something.... If anyone could point out what to do I would appreciate it.
2. (Original post by Square)
having a bit of bother proving the following:

For each positive integer n, let a_n be a real number with a_n[-11,23] and define the sequenceb by

show the sequence is cauchy.

only a 4 mark question on a paper of ~100 marks so it cant be that intensive a question, I think I'm just missing something.... If anyone could point out what to do I would appreciate it.

See anything?
Edit-forgot to add, it should be a less than or equals sign, but I don't know how to do that...
3. If is a positive Cauchy sequence and then b_n is Cauchy. (You should be able to prove this easily).
4. (Original post by Slumpy)

See anything?
Edit-forgot to add, it should be a less than or equals sign, but I don't know how to do that...
no I got this, I'm just getting confused with all the a_n's what do I do with them?
5. (Original post by Square)
no I got this, I'm just getting confused with all the a_n's what do I do with them?
Look at DFranklin's hint above.
You're trying to show that the get closer and closer together, so a way to show that would be to show that the terms being added on are getting sufficiently smaller.
6. i still dont follow, im pretty bad at this!

Here is my working so far.

epsilon>0, N natural number s.t n,m>N

7. (Original post by Slumpy)
Look at DFranklin's hint above.
You're trying to show that the get closer and closer together, so a way to show that would be to show that the terms being added on are getting sufficiently smaller.
quoting so you see this hopefully, check my post above!
8. You know converges, so it's Cauchy. Now compare a multiple of this with your series.
9. (Original post by Square)
i still dont follow, im pretty bad at this!

Here is my working so far.

epsilon>0, N natural number s.t n,m>N

That's you basically there really.
If you prove the result Dave suggested though, you could consider the sequence where a is the largest value can take.
10. I guess this is where the interval on an comes in? So would I want to consider 24a_n (a_n=sum(1/n^2)), given that would be larger than b_n (the sequence above) for all n?
11. (Original post by Square)
I guess this is where the interval on an comes in? So would I want to consider 24a_n (a_n=sum(1/n^2)), given that would be larger than b_n (the sequence above) for all n?
That would work, yes.
12. (Original post by Slumpy)
That would work, yes.
well i really made a meal of that, thanks for the help guys! rep headed your way.
13. (Original post by Square)
well i really made a meal of that, thanks for the help guys! rep headed your way.
No problem, helped me avoid my own work for at least a couple of minutes

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