Prove that $\{ x_n \}^\infty_{i=0}$ where $x_n = r^n$ is a Cauchy sequence (where $0<r<1$)

Question

I do understand how to prove that it is a Cauchy sequence, but I do not know how to apply the method to this sequence.

Answer 1

If $n>m$ then $0 < x_m-x_n =r^m-r^n =r^m(1-r^{n-m}) <r^m$.

Therefore all that is needed to show $r^m \to 0$,

I read this in "What is Mathematics" years ago:

Since $0<r<1$, let $r=1/(1+a)$ where $a>0$. By Bernoulli's inequality, $(1+a)^m > 1+ma >ma$ so $r^m < 1/(ma) \to 0$.