Inspired by Chew-Seong Cheong

Jean can show that a certain algebraic expression satisfies $f(x,y) \geq 4.5$ for all real values of $x$ and $y$ .

The only information we get from the above statement is that 4.5 is a lower bound of $f(x,y)$ , or in other words, the infimum of $f(x,y)$ is greater than or equal to 4.5 . We get no other information whatsoever.

The maximum integer $N$ such that $f(x,y) \geq N$ for all real values of $x$ and $y$ is $\lfloor \inf f(x, y) \rfloor$ . Since all we know is that $\inf f(x,y) \geq 4.5$ , we can only say $N \geq 4$ .

Since there is no largest integer in the interval $[4, \infty)$ , $\boxed{\text{No answer }}$ to this problem exists. $_\square$

Given any $N$ , define $f(x,y)=N+\delta$ with $\delta>0$ being an arbitrarily chosen positive constant.

Then, $f(x,y)>N,\forall x,y$