I have a general question about the way lim sup is usually defined.(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Limit superior: definition and notation

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