Minima of a function

Let $F:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}.$ Is $\inf_{x,y} F(x,y)$ always equal to $\inf_{x} \inf_{y} F(x,y)$?

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It is obvious that $inf_{x,y} F(x,y) \leq \inf_{x} \inf{y} F(x,y).$ –  Lal Dec 25 '12 at 7:44

1 Answer

Yes.

$F(x,y) \geq \inf_y F(x,y) \geq \inf_x \inf_y F(x,y)$, for all $x,y$. Hence $\inf_{x,y} F(x,y) \geq \inf_x \inf_y F(x,y)$.

Similarly, $F(x,y) \geq \inf_{x,y} F(x,y)$, hence $\inf_y F(x,y) \geq \inf_{x,y} F(x,y)$, and hence $\inf_x \inf_y F(x,y) \geq \inf_{x,y} F(x,y)$.

So they are equal...

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