Is it true that for uncountable sets $A,B$ and real valued function $f(\;\cdot\;,\;\cdot\;)$?
$$\inf_{a\in A}\inf_{b\in B} f(a,b)=\inf_{(a,b)\in A\times B} f(a,b) = \inf_{b\in B}\inf_{a\in A} f(a,b).\label{result1}$$ I understand from this question that it is true when both $A,B$ are countably infinite.
How does one deal with if both are uncountable?
This is how much I could proceed: Using $$\inf_{b'\in B} f(a,b') \leq f(a,b)$$ one can obtain
$$\inf_{a\in A}\inf_{b\in B} f(a,b)\leq \inf_{(a,b)\in A\times B} f(a,b)$$
Now I have to prove the opposite of this inequality, and I am stuck at this step.
Can someone point to how this can be done or is it that in general the result is false for uncountable sets?