I will give a proof for the highlighted part, because I don't have any idea how to prove the first part yet, but I will write if I can find.
Originally Posted by hjortur
Using the recursion formula
where and , we learn that
....for all , i.e., is bounded.
Taking now inferior and superior limits on both sides of (1), we obtain
....and.... (actually they are equal, but I prefer inequalities for a general proof),
Set for , and note that is increasing on .
From (3), we have
It follows from (2) and (4) that , i.e., has a finite limit at infinity (more precisely this limit value lies in the interval ).
Taking limit on both sides of (1), we obtain
and solving this we obtain
(note that we took the positive solution of (5) since the solutions we consider are positive),
which is called the equilibrium of the recursive equation (1).
Every solution of (1) with positive initial value and positive , tends to the equilibrium .
You should refer to the following books for rational difference equations:
 M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Champman & Hall, 2002.
 E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Champman & Hall, 2008.
Mathematica codes for plotting a graphic of the solution of the rational sequence
s=30; "Number of iterations";
x[]=1; "Initial Value";