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I want to prove that (without using Fatou's lemma)

for every $k \in N$ let $f_k$ be a nonnegative sequence $f_k(1),f_k(2),\ldots$

$$\sum^\infty_{n=1}\liminf_{k \to \infty} f_k(n) \le \liminf_{k \to \infty} \sum^\infty_{n=1}f_k(n)$$

Can you give some hint for me about that? hat

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Thank you for help. I got this question. Then I wonder why this inequality do for only liminf not for limsup either? –  japee May 8 '12 at 17:32

1 Answer 1

up vote 4 down vote accepted

Hints:

  • Fix an integer $N$, and show that $$\sum_{n=1}^N\liminf_{k\to +\infty}f_k(n)\leq \liminf_{k\to +\infty}\sum_{n=1}^Nf_k(n).$$
  • Show that $\sum_{n=1}^Nf_k(n)\leq \sum_{n=1}^{+\infty}f_k(n)$.
  • Conclude, still using that all the terms are non-negative.
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