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## Homework Statement

Let [tex]X=(x_n) [/tex] be a sequence of strictly positive numbers such that [tex]\lim(x_{n+1}/x_n)<1[/tex]. Show for some [tex]0<r<1[/tex], and for some [tex]C>0[/tex], [tex]0<x_n<Cr^n[/tex]

## Homework Equations

## The Attempt at a Solution

Let [tex]\lim(x_{n+1}/x_n)=x<1[/tex]

By definition of the limit, [tex]\lim(x_{n+1}/x_n)=x \Rightarrow \forall \epsilon>0 [/tex] there exists [tex] \: K(\epsilon) [/tex] such that [tex]. \: \forall n>K(\epsilon) [/tex]

[tex]|\frac{x_{n+1}}{x_n}-x|<\epsilon[/tex]

Since i can pick any epsilon, let epsilon be such that [tex] \epsilon + x = r <1[/tex]. Also, I know that since this is a positive sequence, [tex]\frac{x_{n+1}}{x_n}>0[/tex]. Therefore, for large enough [tex]n[/tex],

[tex]0<\frac{x_{n+1}}{x_n}<r<1. [/tex]

From here I am not sure where to go, any hints would be much appreciated! I cannot find out what this tells me about $x_n$