Understanding Fatou's lemma 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

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.

