# Thread: Cauchy sequence

1. ## 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)}

2. 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)}
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.

3. T is a convex contraction of order 2 means
1)d(T²x,T²y)≤ad(Tx,Ty)+bd(x,y)
2)a,b and a+b∈(0,1)

4. Originally Posted by math3000
T is a convex contraction of order 2 means
1)d(T²x,T²y)≤ad(Tx,Ty)+bd(x,y)
2)a,b and a+b∈(0,1)
Ok, now that we have our notation down, what have you tried?

5. 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(T³x0,T²x0)≤ad(T²x0,Tx0)+bd(Tx0,x0) ≤(a+b)k
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
d(T³x0,T⁴x0)≤ad(T³x0,T²x0)+bd(T²x0,Tx0) ≤(a+b)k
d(T5x0,T6x)≤ad(T⁴x0,T5x0)+bd(T⁴x0,T³x0) ≤(a+b)²k and so on.
Then I arrived to
d(T^{2m+1}x0,T^{2m}x0)≤k(a+b)^{m} and also
d(T^{2m-1}x0,T^{2m}x0)≤k(a+b)^{m}
I want to show that d(T^{m+1}x0,T^{m}x0)≤ A value
then I use it to sow that
∀m<n d(T^{m}x0,Tⁿx0)≤d(T^{m}x0,T^{m+1}x0)+d(T^{m+1}x0,T ^{m+2}x0)+...+d(Tⁿ-¹x0,Tⁿx0)

I want a hint
because I am trying every thing and there is no clearly answer