Limit superior: definition and notation

[kingwinner]

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
k->∞
=lim b_k
k->∞
Here, my understanding is that the indices n and k are independent and are totally unrelated.

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
n->∞
= lim b_n
n->∞
i.e. they replaced n by m in the original sequence and use the same subscript "n" (i.e. b_n), but "n" is already a subscript in the original sequence (a_n), so they can't be independent?
Is it correct to do this and use the same letter n? If so, what is the reason of doing this? Why not use a different index (i.e. a_n and b_k) to show the independence?

Thanks for clarifying!

 

[mathman]

As you observed, the actual definition is the expression on the left side, which is essentially the same in both cases. I can't tell what either source means for the expressions in the middle or the right, since limsup will have no index.

 

[kingwinner]

Sorry, I've made a mistake. It's edited and corrected it now. b_k or b_n is the sequence of supremums of the tails. lim sup is the limit of b_k or b_n.

Can you explain why the expressions on the left sides are the same in both cases?

In the first case, we're taking the limit of a sequence indexed by k. (this makes perfect sense to me because the indices in a_n and b_k should be independent.)

In the second case, we're taking the limit of a sequence indexed by n (which is the exact same index used in the original sequence (a_n), they sneakily replaced a_n by a_m in the definition of lim sup and take m≥n so the resulting seqeunce b_n is indexed by n again)

 

[mathman]

The only difference is the change of letters being used for subscripts. In the first expression we have n≥k and k->∞, while for the second n is replaced by m and k is replaced by n. There is no difference in meaning.

 

[kingwinner]

But the index "n" has already been used in the original sequence {a_n}. Is it OK to use n again for the sequence of supremums of the tails?

 

[mathman]

But the index "n" has already been used in the original sequence {a_n}. Is it OK to use n again for the sequence of supremums of the tails?

Yes. It really doesn't matter. The only place where indices have to be handled carefully is in the defining expression
lim[sup{a_n:n≥k}], where k -> ∞. The sequence itself can be labelled with any index you want.