If $ \ m=\inf \{f(x,y): \ x \in D \} \ $ and $ \ D=[a,b] \times [c,d] \ $

Question

If $ \ m=\inf \{f(x,y): \ x,y \in D \} \ $ and $ \ D=[a,b] \times [c,d] \ $ . Then prove that,

$ \inf \{f(x,y): \ x \in [a,b] \} \geq m \ $ for every $ \ y \in [c,d] \ $

Answer:

We know,

$ \inf \{f(x,y): \ x,y \in D \} \leq \inf \{f(x,y): \ x \in [a,b] , \ y \ \ is \ \ free \} \ $

$ \\ \Rightarrow m \leq \inf \{f(x,y): x \in [a,b] , \ y \in [c,d] \} $

Am I right so far ?

Help me out

Answer 1

$y$ is free

is a bit fishy, because $y$ isn't "free", but fixed! It is much easier to argue like Hagen von Eitzen wrote:

Define $A_y=\{f(x,y)~:~x\in[a,b]\}$ and $A=\{f(x,y)~:~x\in[a,b],~y\in[c,d]\}=\bigcup_{y\in[c,d]}A_y$.

So $A_y\subseteq A$ for all $y\in[c,d]$. Hence $\inf A_y\geq \inf A$ for all $y\in[c,d]$.