QUESTION: show that the sequence http://latex.codecogs.com/png.latex?(x_n) defined recursively as http://latex.codecogs.com/png.latex?... \, unbounded.

MY SOLUTION:

Let the sequence be bounded above. It then has a least upper bound say http://latex.codecogs.com/png.latex?L.

now,

http://latex.codecogs.com/png.latex?...-Lx_n+1 \leq 0

hence:

http://latex.codecogs.com/png.latex?...-4})/2) \leq 0

It can be computed that http://latex.codecogs.com/png.latex?...>4 \, so\, L>4

so we have:

http://latex.codecogs.com/png.latex?...2+\sqrt{L^2-4}

This is true for all http://latex.codecogs.com/png.latex?...\mathbb{Z}^{+} so we have:

http://latex.codecogs.com/png.latex?... 5L^2 \leq -16 Contradiction.

Is this correct?

also if you have a good method to do it then please post it.