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, ak> ak+1. Then ak+1+1= (ak+1+1)/ak+1+ 2). Again you have that but, as you say, since both numerator and denominator are larger than in ak+1, that doesn't tell you anything. Perhaps it would help to recognise that (x+ 1)/(x+ 2)= 1- 1/(x+ 2). If uk+1< uk then uk+1+ 2< uk+ 2 so 1/(uk+1+ 2)> 1/uk and then -1/(uk+1)< -1/uk.
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!
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 cant understand how you got them
and how i go further to proove my inequality
can you please wright me the solution to this problem