By trying to extend certain limit properties of sequences from compact subsets to the entire set, I cam up with something that can be formed in the following question.


Let $a\_{mn}$ be a double sequence of nonnegative real numbers. I want to be able to switch order of iterated limits in the following form
$$\limsup\_{m \to \infty}\limsup\_{n \to \infty} a\_{mn}=\limsup\_{n \to \infty}\limsup\_{m \to \infty} a\_{mn}$$
under the following conditions


* $a\_{mn}$ is increasing in $m$ for every $n$
* $\limsup\_{m \to \infty} a\_{mn}=\lim\_{m \to \infty} a\_{mn}=a\_{n}, \forall n$
* $\limsup\_{n \to \infty} a\_{mn} \leq a\_m, \forall m$ where $a\_m$ is increasing in $m$


I tried all of the ideas from [here](https://math.stackexchange.com/questions/15240/when-can-you-switch-the-order-of-limits), but without any success. Any helpful ideas or references?



Switching order of limits in double sequences
$a\_{mn}=1(m\ge n)$ is a counterexample. (Here, $1(A)$ is the indicator of an assertion $A$. That is, $1(A):=1$ if $A$ is true and $1(A):=0$ if $A$ is false.)


If you insist on understanding "increasing" in the strict sense, then $a\_{mn}=1(m\ge n)-1/m-1/n$ is a counterexample.


