# Cauchy sequence

1. Jan 27, 2014

### dirk_mec1

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution
I've tried different approaches (including induction) but the sequence isn't monotonically decreasing.

2. Jan 27, 2014

### Office_Shredder

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

3. Jan 27, 2014

Got it ty.