$a_{mn}=1(m\ge n)$ 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.