Cauchy Sequence Homework: Show x_n is Cauchy

[dirk_mec1]

Homework Statement

Given:

$$x_{n+1}=\frac{1}{3+x_n}$$

with
$$x_1=1$$

Show that:

(1)

$$|x_{n+1}-x_n| \leq \frac{1}{9}|x_{n}-x_{n-1}|$$

and (2) x_n is Cauchy.

Homework Equations

The Attempt at a Solution

I've tried different approaches (including induction) but the sequence isn't monotonically decreasing.

 

[Office_Shredder]

Have you tried writing out $|x_{n+1} - x_{n}|$ using the definition $x_j = 1/(3+x_{j-1}$?

 

[dirk_mec1]

Got it ty.