This was on an Analysis I short test I took yesterday:
$K$ is an ordered field, let $M \subset K$ and $x,y \in K$ with $|x|<|y|$ so that
$$M=\{z\cdot z :x<z\leq|y| \}$$
For which $x$ and $y$ does a minimum or supremum exist in $K$? Calculate them if they exist.
My answer:
The minimum is $0$ for $x \in (-P)$ and for all $y$. $0 \in K$.
The infimum is $x\cdot x$ for $x \in \{0\}\cup P$ and for all $y$. Since $x\cdot x \notin M$ there exists no minimum.
The supremum is $y \cdot y$ for all $x$ and $y$ and since K is a field and $y \in K$, $y\cdot y \in K$.
Is my answer correct? The way I went on proving this was dubious at best and I would like to know how it could be done properly or if there are any general tips that help prove such things.