A sequence $\displaystyle \left \{x_n\right \}_{n\ge1}$ is called contractive iff there exists a $\displaystyle c\in \left.[0,1 \right )$ such that, for all $\displaystyle n\ge1$

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

Any contractive sequence is Cauchy.

Use the above definition to prove that the sequence defined by

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

is convergent and evaluate its limit.

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

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

SOLVED

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