Prove cauchy sequence and thus convergence

[manooba]

Let (Xn) be a sequence satisfying

|Xn+1-Xn| ≤ λ^n r

Where r>0 and λ lies between (0,1). Prove that (Xn) is a Cauchy sequence and so is convergent.

 

[radou]

I have a hunch that you could use two facts:

1) for every ε > 0 there exists some natural number N such that λ^N r < ε
2) the triangle inequality

 

[tim e]

Let (Xn) be a sequence satisfying

|Xn+1-Xn| ≤ λ^n r

Where r>0 and λ lies between (0,1). Prove that (Xn) is a Cauchy sequence and so is convergent.

I posted the solution of that with r=1 in a similar thread called cauchy sequence problem