Limit superior: definition and notation

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  • #1
kingwinner
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I have a general question about the way lim sup is usually defined.

Let (an) be a sequence of real numbers. Then we define lim sup to be
lim [sup{an: n≥k}] = lim sup an
k->∞
=lim bk
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 (an) be a sequence of real numbers. Then they define lim sup to be
lim [sup{am: m≥n}] = lim sup an
n->∞
= lim bn
n->∞
i.e. they replaced n by m in the original sequence and use the same subscript "n" (i.e. bn), but "n" is already a subscript in the original sequence (an), 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. an and bk) to show the independence?

Thanks for clarifying!
 
Last edited:

Answers and Replies

  • #2
mathman
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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.
 
  • #3
kingwinner
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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)
 
  • #4
mathman
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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.
 
Last edited:
  • #5
kingwinner
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But the index "n" has already been used in the original sequence {an}. Is it OK to use n again for the sequence of supremums of the tails?
 
  • #6
mathman
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But the index "n" has already been used in the original sequence {an}. 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{an:n≥k}], where k -> ∞. The sequence itself can be labelled with any index you want.
 

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