I have a sequence of real numbers for which I wish to prove convergence.

and

So, if the sequence converges to a limit then

Also, we have that

But I have yet to prove convergence.

So if and are both less than , then and our initial conditions give us .

And thus the sequence is bounded above.

So now I need to prove that sequence is increasing.

But that doesn't show that as I require.

Can you help?