Context
I am reading about Convex Analysis and more specifically the famous Minimax Theorem. It goes on about what to do if the function is convex, strictly convex, concave and strictly concave. All of these I understand. However one piece of this is missing.
In the last paragraph, it says:
Finally, the claim for the last equality in the formulation of theorem can be gotten from the following equality and taking note that $\inf \to \min$ and $\sup \to \max $.
The "equality" in question is (I am citing the full theorem)
If \begin{align} (x_0, y_0) \in X \times B \text{ such that } K(x, y_0) \leq K(x_0, y_0) \leq K(x_0, y) \end{align} then $(x_0, y_0)$ saddle point and then we can write \begin{align} \sup_{x \in A} \inf_{y \in B} K(x,y) = \inf_{y \in B} \sup_{x \in A} K(x,y) \end{align}
So basically what the author says is that by taking the aforementioned equality and putting $\min$ where $\inf$ and $\max$ where $\sup$ we can get the Minimax Theorem. My question is? How exactly? A function satisfying the inequality means that $(x_0, y_0)$ is saddle point. How can that translate to the equality?
I am not even sure if I put my thoughts correctly, since my learning book is quite poor in overall quality.
I am expecting for us to work through it.
Suppose $K(x,y_0) \le K(x_0, y_0) \le K(x_0, y)$
It is always true that $\sup_x \inf_y K(x,y) \le \inf_y \sup_x K(x,y)$.
From the above we have $K(x,y_0) \le K(x_0, y_0) \le \inf_y K(x_0, y)$
$\sup_x K(x,y_0) \le K(x_0, y_0) \le \inf_y K(x_0, y)$
$\sup_x K(x,y_0) \le K(x_0, y_0) \le \inf_y K(x_0, y) \le \sup_x \inf_y K(x, y)$
$\inf_y \sup_x K(x,y) \le \sup_x K(x,y_0) \le K(x_0, y_0) \le \inf_y K(x_0, y) \le \sup_x \inf_y K(x, y)$