If you take , we don't have so we can't deduce the conclusion of the result is true.
The result is surely true because a convergent sequence is bounded.
I must demonstrate that if , and are sequences of real number then the following is true:
if then .
My attempt: I must show that knowing that then there exist a constant and an integer such that for .
But to me, any constant in R that I'd take would make . For instance if the sequence , then . In fact C would have to be infinite for the relation to be true.
Am I missing something or the problem asks me to show a wrong result?
Thanks Jose. Actually I think it would be overkill.
I want to correct my previous post into this one: The existance of a constant C and an integer r such that for n>r is equivalent to write .
But I start with knowing that and since then the implication is obvious.
I said overkill in your base because from what I understand you want to prove it for all n while I think they ask for , unless I'm misunderstanding once again the exercise.
They do not state "for sufficiently large n", but this is what I believe from the definitions I have for the little and big o notations. The problem is the one I wrote in the first post in the lines before "My attempt".
By the way I didn't know the result would hold for any n, I think this is a nice result and I thank you for pointing it out.