# Prove this sequence is contractive and evaluate its limit

• January 27th 2013, 05:05 AM
quepux
Prove this sequence is contractive and evaluate its limit
A sequence $\left \{x_n\right \}_{n\ge1}$ is called contractive iff there exists a $c\in \left.[0,1 \right )$ such that, for all $n\ge1$

$\left |x_{n+2}-x_{n+1}\right| \ge c\left|x_{n+1}-x_{n}\right |$

Any contractive sequence is Cauchy.

Use the above definition to prove that the sequence defined by

$x_1 = \alpha > 2, x_{n+1}=\displaystyle{x_n+2 \over x_n}$

is convergent and evaluate its limit.

$\left |x_{n+2}-x_{n+1}\right|=\displaystyle{2 \over x_{n+1}x_{n}}\left|x_{n+1}-x_{n}\right|$

I couldn't go any further than this, any help will be appreciated.

SOLVED

$x_{n+1}x_n=x_n+2$ then by induction $x_n>1$