Analysis of "X = (xn) Sequence of Strictly Positive Real Numbers

[buzzmath]

X = (xn) is a sequence of strictly positive real numbers, where (xn) is x subscript n, such that lim(x(n+1)/xn) < 1. Show that for some r with 0<r<1 and some C>0 that 0<xn<Cr^n for all sufficiently large natural numbers n. and that lim(xn) = 0

So for I have this:
choose r such that lim(x(n+1)/xn)<r<1 and take a neighborhood of this limit to be the interval (-1,r) So there exists a natural number K such that 0<x(n+1)/xn<r for all n>=K. I can also write r = 1/(1+a) where a>0 and show that lim(r^n)=0. All I need to show now is that xn<Cr^n. Because I know that if lim(r^n)=0 and ||xn - 0||<=C|r^n| where C>0 then lim(r^n)=0. I'm not really sure how to get the xn<Cr^n though. Any help or suggestions?

 

[StatusX]

You've shown x_n+1/x_n<r for n>=K. You can rewrite this as x_n+1<r*x_n. Apply induction to get an inequality involving x_k+n and x_k.