- #1

- 1,270

- 0

I have a general question about the way lim sup is usually defined.

Let (a

lim [sup{a

k->∞

=lim b

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

lim [sup{a

n->∞

= lim b

n->∞

i.e. they replaced n by m in the original sequence and use the same subscript "n" (i.e. b

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

Thanks for clarifying!

Let (a

_{n}) be a sequence of real numbers. Then we define lim sup to belim [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 belim [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!

Last edited: