# Cauchy sequence

• Apr 29th 2010, 12:13 PM
math3000
Cauchy sequence
Hello, everyone Iwant to ask about Cauchy seq.
How to prove that d(x2m+1,x2m)and d(x2m-1,x2m) are cauchy sequences
if T:X->X is a convex contraction mapping of order 2 , X is complete metric space ,xn+1=Txn and k=max{d(x1,x0),d(x1,x2)}
(Hi)
• Apr 29th 2010, 12:15 PM
Drexel28
Quote:

Originally Posted by math3000
Hello, everyone Iwant to ask about Cauchy seq.
How to prove that d(x2m+1,x2m)and d(x2m-1,x2m) are cauchy sequences
if T:X->X is a convex contraction mapping of order 2 , X is complete metric space ,xn+1=Txn and k=max{d(x1,x0),d(x1,x2)}
(Hi)

This is hard to read. Could you try to write it in Latex? I'm not sure what a "convex" contract is or what "order 2" means.
• Apr 29th 2010, 12:43 PM
math3000
T is a convex contraction of order 2 means
2)a,b and a+b∈(0,1)
• Apr 29th 2010, 09:03 PM
Drexel28
Quote:

Originally Posted by math3000
T is a convex contraction of order 2 means
2)a,b and a+b∈(0,1)

Ok, now that we have our notation down, what have you tried?
• Apr 30th 2010, 01:38 AM
math3000
Quote:

Originally Posted by Drexel28
Ok, now that we have our notation down, what have you tried?

From the inequality
d(T^{2m+1}x0,T^{2m}x0)≤ad(T^{2m}x0,T^{2m-1}x0)+bd(T^{2m-1}x0,T^{2m-2}x0) and for m≥1 .I get
d(T5x0,T⁴x0)≤ad(T⁴x0,T³x0)+bd(T³x0,T²x0) ≤(a+b)²k and so on.
and From the inequality
d(T^{2m-1}x0,T^{2m}x0)≤ad(T^{2m-2}x0,T^{2m-1}x0)+bd(T^{2m-3}x0,T^{2m-2}x0) and for m≥2 .I get