Is this discrete analogue of Fatou's lemma valid I was curious if this discrete analog of Fatou's lemma is valid:

\begin{align}
\sum_{j=1}^\infty \liminf_{k \rightarrow\infty} a_j(k) \leq \liminf_{k \rightarrow\infty} \sum_{j=1}^\infty a_j(k) ,
\end{align}
where $a_j(k)$ is a doubly indexed sequence of real numbers.

Does it hold in the general real case? What if $a_j(k) \geq 0 $, does it hold then? Thanks to all helpers.

An infinite sum can be realized as an integral w.r.t counting measure on the integers. So the inequality holds in the non-negative case. Fatou's Lemma requires non-negativity and and the inequality is false without it.
$a_j(k)=-1$ for $j=k$ and $0$ for $j \neq k$ gives a  counter-example.
