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$\begingroup$

$$\sum^N_{n=1}\liminf_{k \to \infty} f_k(n) = \lim_{k \to \infty} \sum_{n=1}^N \inf_{j \ge k} f_j(n)$$

I am not sure that equation true. Is that equation true? Then why is it?

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    Why don't you specify what are those objects you're working with?2012-05-08
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    I just wanna solve http://math.stackexchange.com/q/142728/30883 problem. But I have no basic knowledge wanna have.2012-05-08

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It is known that for any sequence $\{a_k:k\in\mathbb{N}\}\subset\mathbb{R}$ we have $$ \liminf\limits_{k\to\infty}a_k=\lim\limits_{k\to\infty}\inf\limits_{j\geq k}a_j $$ so $$ \sum\limits_{n=1}^N\liminf\limits_{k\to\infty}f_k(n)= \sum\limits_{n=1}^N\lim\limits_{k\to\infty}\inf\limits_{j\geq k}f_j(n)= \lim\limits_{k\to\infty}\sum\limits_{n=1}^N\inf\limits_{j\geq k}f_j(n) $$

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    Oh thanks. Is last equation undoubtfully true?(switching sum and limit)2012-05-08
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    It is true, because sum is finite and doesn't depend on limit's variable.2012-05-08
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    Oh I got it. Thanks @norbert!2012-05-08
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    @japee Not at all!2012-05-08
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    @japee By the way, you can accept this answer and answers for other your questions - click grey checkmark under two vertical arrows2012-05-08
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    Ah I clicked and it change to green button. I got it!!2012-05-08
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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/3373/discussion-between-japee-and-norbert)2012-05-08
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    Sure! I accepted other three now. I got scholar badge indeed. Thanks!2012-05-08