In strict lexicographical ordering :Given the following
Find, if possible, three explicit upper bounds for $C = \{(x, y) \in \mathbb{R}^2 | y < 0\}$. and write the set of upper bounds for $C$ in proper set notation. Does $C$ have a least upper bound? No proof is necessary, but do write a sentence or two explaining why or why not.
I had that the set of upper bounds would be $\{(x,y) \in \mathbb{R}^2 | y \geq 0\} $ and three examples of upper bounds would be (-1,5),(3,4), and (-5,10). I found out my answer was incorrect, but I was confused as to why and where I went wrong - any hints or suggestions would be appreciated.
Also, I'm not sure if $C$ has a least upper bound or not, and I'm not sure how to figure it out.
In lexicographic ordering, $(x_1,y_1)<(x_2,y_2)$ if and only if either
Since the elements of $C$ range over all possible $x$-values, then it has no upper bounds. Indeed, if $(x^*,y^*)$ were an upper bound of $C$, then for all $x,y\in\Bbb R$ with $y<0,$ we would have $x<x^*$ or $x=x^*$ and $y<y^*.$ Consequently, $x\le x^*$ for all $x\in\Bbb R,$ so since $x^*\in\Bbb R,$ this would mean that $x^*$ is the greatest element of $\Bbb R.$