Originally Posted by
kingwinner OK, then I think I understand your proof. Thank you!
But now I have a general question about the way lim sup is usually defined.
Let (a_n) be a sequence of real numbers. Then we define lim sup to be
lim [sup{a_n: n≥k}] = lim sup a_n = b_k
k->∞
Here, my understanding is that the indices n and k are independent.
But I have seen some textbooks doing the following:
Let (a_n) be a sequence of real numbers. Then they define lim sup to be
lim [sup{a_m: m≥n}] = lim sup a_n = b_n
n->∞
i.e. they are using the same subscript "n", but "n" is the subscript in the original sequence (a_n), so they can't be independent.
Is it correct to do this?
If we use the latter definition, would this ruin our proof? (becuase at the end, I don't think we'll get the same conclusion...)