Here is my question. Let $a_{m,n}$ be a positive sequence, and I have that $a_{m,n}\leq L<+\infty$ for all $n$ and $m$. I also know that $\lim_{m\to \infty}a_{m,n}= a_n$ for each $n$ fixed. I understand that if I have $$ \lim_{m\to \infty}\sup_{n\geq 0}|a_{m,n}-a_n|=0,\tag 1 $$ then I have $$ \liminf_{m\to \infty}\liminf_{n\to \infty}a_{m,n} = \liminf_{n\to \infty}\liminf_{m\to \infty}a_{m,n} \tag 2 $$ However, I do not have condition (1) but luckily I only need the half strength of (2), i.e., I only want $$ \liminf_{m\to \infty}\liminf_{n\to \infty}a_{m,n} \geq \liminf_{n\to \infty}\liminf_{m\to \infty}a_{m,n} \tag 3 $$ So, would $(3)$ hold without $(1)$? Or, in the case it does not, can I get a weaker condition to ensure that $(3)$ hold?
Update: from the answer below we know that $(3)$ does not hold in general. So, I update $(1)$ to be $$ \lim_{m\to \infty}\sup_{n\geq 0}(a_n-a_{m,n})\leq0,\tag 4 $$ That is, if $(4)$ hold, we have $(3)$. Here is a quick prove:
We have $$ \liminf_{n\to\infty}\lim_{m\to\infty}a_{m,n}=\liminf_{n\to\infty}a_n $$ by the fact that $\lim_{m\to \infty}a_{m,n}=a_n$.
Moreover, by $(4)$ we have $$ a_n\leq a_{m,n}+c_m $$ for all $n$ where $c_m\to 0$ as $m\to\infty$. Hence, we have $$ \liminf_{m\to\infty}\liminf_{n\to\infty}a_{m,n}\geq \liminf_{n\to\infty}a_n $$ and we are done.
