- #1
Dollydaggerxo
- 62
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Homework Statement
Well, my problem is proving that sequences are in fact Cauchy sequences. I know all the conditions that need to be satisfied yet I cannot seem to apply it to questions. (Well, only the easy ones!)
My question is, prove that [tex]X_{n}[/tex] is a Cauchy sequence, given that [tex]X_{n+1}= f(X_{n})[/tex] where f(x)=cos(x).
I have been told that [tex]|X_{n+1}-X_{n}| \leq 2^{(1-n)}[/tex]
The attempt at a solution
Well basically, I tried using m,n>N and N>0 and m>n>N to say that
[tex]|X_m-X_n| = cos(X_{m-1}) - cos(X_{n-1})[/tex]
because this is the way i was doing it for simpler sequences.
However, I haven't got a clue where I would go from here which leads me to thinking I have gone the wrong way about it!
Any help would be greatly appreciated.
Thanks
Well, my problem is proving that sequences are in fact Cauchy sequences. I know all the conditions that need to be satisfied yet I cannot seem to apply it to questions. (Well, only the easy ones!)
My question is, prove that [tex]X_{n}[/tex] is a Cauchy sequence, given that [tex]X_{n+1}= f(X_{n})[/tex] where f(x)=cos(x).
I have been told that [tex]|X_{n+1}-X_{n}| \leq 2^{(1-n)}[/tex]
The attempt at a solution
Well basically, I tried using m,n>N and N>0 and m>n>N to say that
[tex]|X_m-X_n| = cos(X_{m-1}) - cos(X_{n-1})[/tex]
because this is the way i was doing it for simpler sequences.
However, I haven't got a clue where I would go from here which leads me to thinking I have gone the wrong way about it!
Any help would be greatly appreciated.
Thanks
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