Fatou's Lemma for sequences

Question

Is there some kind of Fatou's Lemma for Sequences along the following lines:

Suppose I have a doubly indexed array $\{a_{n,k}\}_{n,k =1}^{\infty}$ and I know the following facts:

1. For each fixed $n$, $\liminf_{k \to \infty}a_{n,k} \geq a_n$.
2. For each fixed $k$, the sequence $\{a_{n,k}\}_{n=1}^{\infty}$ is monotone (let's say decreasing).
3. $\{a_n\}_{n=1}^{\infty}$ is also monotone decreasing.

Then, can I conclude something like: $$ \lim_{n \to \infty}\liminf_{k \to \infty}a_{n,k} \leq \liminf_{k \to \infty}\lim_{n \to \infty}a_{n,k}? $$ The limit in $n$ is playing the role of the Lebesgue integral (which is a type of monotone limit). I think Hypothesis 1. is kind of redundant, but I'll leave it there anyway.

Answer 1

Counterexample: $a_{n,k}=2^{k-n}$, $a_n=0$.

As $k\to\infty$, $a_{n,k}\to\infty$, so $\liminf_k a_{n,k} = \infty > a_n$. $$\lim_{n\to\infty}\liminf_{k\to\infty}a_{n,k} = \lim_{n\to\infty}\infty=\infty$$ $$\liminf_{k\to\infty}\lim_{n\to\infty}a_{n,k} = \liminf_{k\to 0}0=0$$ That inequality points in the wrong direction.

There might be something to salvage, but it'll take reversing some inequality signs in the conditions.