If the 'recursive relation' is...

(1)

... it can be written as...

(2)

Now for n 'large enough' is with good approximation...

(3)

... and f(x) is represented here...

The f(x) has an 'attractive fixed point' at and a 'repulsive fixed point' at so that for the sequence will converge monotonically at 2...

Kind regards