I have a quick notational question.
Would correct logical notation for suprema and infima of some set $A \subset \mathbb{R}$ be:
$$\sup A = s : \forall a \in A : \forall b \ge a : a \le s \le b$$
$$\inf A = s : \forall a \in A : \forall b \le a : b \le s \le a$$
I could not find results quite answering this on MSE.
And, I am taking an advanced calculus or introduction to analysis course that uses Abbott's Understanding Analysis.
Let $A = [0, 1]$. We have that $\sup A = 1$. However, it is not true that $\forall a \in A : \forall b \geq a : a \leq s \leq b$.
Take, for example, $a = 0, b = \frac{1}{2}$. Both $a \in A$ and $b \geq a$ are true, but it is not the case that $0 \leq 1 \leq \frac{1}{2}$.
You need to encode this quality that $s$ is the least such upper bound, that is for every upper bound $b$, the inequality you have is satisfied. In other words, for every $a \in A$ and every $b$ such that $b \geq c$ for all $c \in A$, $a \leq s \leq b$. Let me know if this helps.