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)}
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