Let $$L = \{(x,y) \in \mathbb R^{2} \mid x ≤ y \}$$
$$B = \{x \in \mathbb R \mid x ≥ 7 \}$$
$$C = \{x \in \mathbb R \mid x > 7 \}$$
Does B have any L-smallest or L-minimal elements? What about the set C?
According to the book I'm reading:
$B$ does have L-smallest and L-minimal element, which is $7$
However, $C$ does not have L-smallest and L-minimal element.
My question is, why? Isn't L-smallest element in $C$ the one that is closest to $7$? I suppose it will look this:
$$7.000000000000.....$$
Where there is a lot, apparently infinitely many zeros in the decimal part and $1$ in the very end, but still, such number must exist, no?
Such a number exists and is called $7$ which does not lie in $C$. Suppose $x\in C$ is the smallest element of $C$. We have that $x>7$, thus $x>y=\frac{1}{2}(x+7)>7$, so $y\in C$ and smaller than $x$, which contradicts $x$ being the smallest element of $C$. So $C$ has no smallest element.