$$\sum^N_{n=1}\liminf_{k \to \infty} f_k(n) = \lim_{k \to \infty} \sum_{n=1}^N \inf_{j \ge k} f_j(n)$$
I am not sure that equation true. Is that equation true? Then why is it?
Asked
Active
Viewed 63 times
1 Answers
3
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 $$ \sum\limits_{n=1}^N\liminf\limits_{k\to\infty}f_k(n)= \sum\limits_{n=1}^N\lim\limits_{k\to\infty}\inf\limits_{j\geq k}f_j(n)= \lim\limits_{k\to\infty}\sum\limits_{n=1}^N\inf\limits_{j\geq k}f_j(n) $$
Norbert
- 56,803
-
-
1It is true, because sum is finite and doesn't depend on limit's variable. – Norbert May 08 '12 at 20:54
-
-
-
@japee By the way, you can accept this answer and answers for other your questions - click grey checkmark under two vertical arrows – Norbert May 08 '12 at 21:35
-
-
-