change the order of sup and inf

Question

I know that for a series $a_{m,n}\ge 0$ $$\sup_m \inf_n a_{m,n}\neq \inf_n\sup_m a_{m,n}$$

But I can't find a counterexample. Could anyone help me?

Thanks a lot!

Answer 1

Thanks to Martin Sleziak's hint:

$a_{m,n}=1_{m>n}+2_{m\le n}$

$$1=\sup_m \inf_n a_{m,n}\neq \inf_n\sup_m a_{m,n}=2$$

Answer 2

This reminded me of the difference between $\limsup$ and $\liminf$, so I thought of $$ a_{m,n}=(-1)^{m+n} $$ where $$ \sup_m\operatorname*{inf\vphantom{p}}_na_{m,n}=-1 $$ and $$ \operatorname*{inf\vphantom{p}}_n\sup_ma_{m,n}=+1 $$