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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!
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!
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