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I am thinking about a sequence such that the following holds (noticed > not $\geq$)

$\liminf_{n \to \infty} (a_n + b_n) > \liminf_{n \to \infty} a_n + \liminf_{n \to \infty} b_n$

I am not sure if this is allowed, but I tried doing something like

$a_n = \left \{2....-1,1,-1,1,-1,1 \right \}$

$b_n = \left \{2....1,-1,1,-1,1, -1\right \}$

I know the example doesn't work, but can you actually write down the limsup (the end term) like this? Or is this as erroneous as writing $1/0$?

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    Sprite sequences are lemon-liming.2012-10-10

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No, what you’ve written for $a_n$ and $b_n$ simply doesn’t make sense. The sequences that you’re to construct are ordinary infinite sequences $\langle a_1,a_2,\dots\rangle$ and $\langle b_1,b_2,\dots\rangle$, with no last elements.

HINT: Take $a_n=(-1)^n$, so that $\liminf_na_n=-1$. Can you find another sequence $\langle b_1,b_2,\dots\rangle$ such that $\liminf_nb_n$ is also $-1$, but $\langle a_1+b_1,a_2+b_2,\dots\rangle$ is constant with a value greater than $-1$?

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    @jak: I’ll rephrase that, since it wasn’t clear. I meant that infinite sequences, which is clearly what we’re talking about here, do not have last elements. *Your sequences* referred not to the things that you wrote down, but to the $\langle a_1,a_2,\dots\rangle$ and $\langle b_1,b_2,\dots\rangle$ that the problem talked about.2012-10-10