I'm trying to prove the following claim:
Let $f, g : [0,1]^2 \to \mathbb{R}$ be continuous functions. Assume that $(x^*,y^*) \in (0,1)$ is the only point satisfying $$\begin{cases}f(x^*, y^*)=\max\limits_{x} f(x,y^*) \\ g(x^*, y^*)=\max\limits_{y} g(x^*,y)\end{cases}$$
Define $$\begin{cases}\alpha := \max\limits_{x \in [0,1]}\,\, \min\limits_{y \in [0,1]} f(x,y) \\ \beta := \max\limits_{y \in [0,1]}\,\,\, \min\limits_{x \in [0,1]} g(x,y)\end{cases}$$
Prove that $\min\limits_{y \in [0,1]} f(x^*,y)=\alpha$ and $\min\limits_{x \in [0,1]} g(x,y^*)=\beta$.
My approach here was to define $k(x)=\min\limits_{y \in [0,1]} f(x,y)$ and $l(y)=\min\limits_{x \in [0,1]} g(x,y)$. $k$, $l$ are continuous, and satisfy $\alpha=\max\limits_x k, \beta=\max\limits_y l$. We want to show that $k(x^*)=\alpha, l(y^*)=\beta$.
I tried each equality by showing the two inequalities $\le, \ge$, and I tried to prove these by contradiction. It unfortunately didn't work.
I think that my problem was that I didn't manage to use the fact the game is over the unit square. Perhaps it may be helpful by defining an infinite sequence and showing convergence. Thanks in advance.