$\sum^N_{n=1}\liminf_{k \to \infty} f_k(n) = \lim_{k \to \infty} \sum_{n=1}^N \inf_{j \ge k} f_j(n)$
I am not sure that equation true. Is that equation true? Then why is it?
$\sum^N_{n=1}\liminf_{k \to \infty} f_k(n) = \lim_{k \to \infty} \sum_{n=1}^N \inf_{j \ge k} f_j(n)$
I am not sure that equation true. Is that equation true? Then why is it?
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) $