Is it possible to write the maximum of a function in the iterated form?

Question

Is the following equality true

$\max_{(u,v)\in C\times D}|f(u,v)|=\max_{u\in C}\max_{v\in D}|f(u,v)|$

From my understanding, the $\max|f(u,v)|$ over a rectangle region shall be the same to the number we get from taking maximum of $|f(u,v)|$ over $v\in D$ for each $u$ fixed, then take maximum over $u\in C$, but I am not quite sure about this.

Answer 1

If $M=|f(u_0,v_0)|$ is the left-hand side. Then for all $u,v$ $M\geq|f(u,v)|$.

This means that for all $u$, $M\geq \max_v|f(u,v)|$.

Therefore, $M\geq\max_u\max_v|f(u,v)|$

But $\max_u\max_v|f(u,v)|\geq \max_v|f(u_0,v)|\geq|f(u_0,v_0)|=M$

Therefore, equality holds.