
induction
The questions asks ...
Suppose that , , and
,
Prove: For , and
MY attempt at a solution:
I really don't know how to go about this... at first i tried to show that if is true then .
So that means
My problem is that i don't know how big R in comparison to , so unless i restrict R to be less than i can't prove that
Basically, i need help with this question... can anyone help ?
I'm doing a self study by myself so i would appreciate a detailed solution if possible.. please and thank u.

I believe you are right, in fact if (for
instance and then the sequence increases in the first terms , but and then the hypothesis seems to hold (for
as stated).
However, if we have the constant sequence
I conjecture without doing any more calculations that this is the unique problem with the hypothesis, we have to require .
I suggest
a) Try to prove the inequality if , it seems an easy induction at least for the first one.
b) Try to show using differential calculus that
maps into . This would solve the case fos "small 's"
because it would allow to apply a)
I haven't made the calculations but it seems to me the way of attacking it.

Sorry i'm not very sure about what you're asking me to do.
My point was that is not true for all n if R> . So do i restrict R so that this case is satistfied ?
If i do this then it would seem that the sequence is decreasing at first but then it would blow up to infinity .
The question is puzzuling to me.

My idea is to divide the problem in the following cases:
a) If then and the sequence tends to . I believe it follows from the inequality you obtained in the first post, by an easy induction. Observe that the inequality you got is . If then you have clearly and you can apply induction. For showing you have to use that is increasing in and .
b) If then for all .
c) If then and then by the same argument of a). The key for obtaining is to observe that
is decreasing in and .
I mean that , has a unique attracting point , the minimum, and that any iteration goes to this point. The case b)
is an exception to your statement since it involves strict inequalities. c) is NOT an exception since you have to show for ,
If it is still not clear I will try to write any step detailedly, but I don't have now the necessary time.