A question in prooving function convergence

[transgalactic]

i showed the question and how i tried to solve it
in the link

http://img293.imageshack.us/my.php?image=img8139an2.jpg

 

[HallsofIvy]

a₀= 1, a_n+1= (a_n+ 1)/(a_n+ 2)

I take it you want to prove that it is decreasing. It's clearly bounded below (by 0), so it has a limit. And then find the limit. a₁= (1+ 1)/(1+ 2)= 2/3< 1. That you have.

Now, suppose, for some k, a_k> a_k+1. Then a_k+1+1= (a_k+1+1)/a_k+1+ 2). Again you have that but, as you say, since both numerator and denominator are larger than in a_k+1, that doesn't tell you anything. Perhaps it would help to recognise that (x+ 1)/(x+ 2)= 1- 1/(x+ 2). If u_k+1< u_k then u_k+1+ 2< u_k+ 2 so 1/(u_k+1+ 2)> 1/u_k and then -1/(u_k+1)< -1/u_k.

You then solve t= (t+1)/(t+2) and get two solutions. Of course, only one of those is the limit of the sequence. The fact that only one of them is positive should make it clear which!

 

[transgalactic]

i tried to use what you told me
i have written your explanation several times
some steps in your post that you say "then"
i can't understand how you got them
and how i go further to proove my inequality

can you please wright me the solution to this problem
??

 

[HallsofIvy]

No, I can't!