Is there any counter-example to $\inf_{x, y}f(x,y) = \inf_{x}\inf_{y}f(x,y)?$

Question

Thanks! I couldn't think of any counter-examples.

Answer 1

By definition of $\inf$ we have $\inf_{x,y}f(x,y)\leq \inf_y f(x,y)$ and therefore $$\inf_{x,y}f(x,y)\leq \inf_x\inf_y f(x,y).$$

On the other hand there is a sequence $x_n,y_n$ such that $f(x_n,y_n)\to\inf_{x,y}f(x,y)$. Therefore $\inf_yf(x_n,y)\leq f(x_n,y_n)$ and therefore $$\inf_{x}\inf_yf(x,y)\leq\inf_x\inf_yf(x_n,y)\leq f(x_n,y_n)$$

Taking limit $\lim_{n\to\infty}$ on both sides we get

$$\inf_x\inf_y f(x,y)\leq\inf_{x,y}f(x,y).$$

Therefore $$\inf_{x,y}f(x,y)=\inf_x\inf_yf(x,y).$$