infimum's basic properties in optimization problem

Question

This problem is in Optimizing over some variables slide of Convex Optimization problem. I have a question about basic assumption in this textbook. $$ \inf_{x,y} f(x,y) = \inf_{x} g(x), where, g(x)=\inf_{y} f(x,y) $$ How can I prove it??

Answer 1

This equality should be obvious. Put $A=\inf_{x,y} f(x,y)$, $B=\inf_{x} g(x)$.

$A\le B$: Let $\varepsilon>0$ be an arbitrary number. Therefore there exists $x$ such that $g(x)<B+\varepsilon$. Hence there exists $y$ such that $f(x,y)<B+\varepsilon$. Then $A\le f(x,y)< B+\varepsilon$.

$B\le A$: Let $\varepsilon>0$ be an arbitrary number. Therefore there exist $x,y$ such that $f(x,y)<A+\varepsilon$. Then $B\le g(x)\le f(x,y)< A+\varepsilon$.