A quick doubt, lets study the recurrence sequence:
First of all i do:
If this is true i can say that decrease. This is true for those values:
And false (so increase) for those values:
the case interests me.
The limit L can be -1 or -2 but i cannot say it exists for sure because is not limited and monotone for all the possible values. For example, the sequence can go from then decrease and go in then again increase and fall in etc...
Another doubt comes from this fact:
It's ok to remove -1 from the possible values of L because in this case the sequence still growing?
Anyways: It happens so many times that i know the sequence increase or decrease in an interval but i don't know if doing it it will fall in another interval where it starts decreasing or increasing and in this scenario i don't know how to demonstrate if it goes on some limit or just starts to "ping-pong" on different intervals.
Or in other words i don't know how many it decrease/increase so i cannot say if it will go out from the interval i'm considering.