Approximation Property of the Supremum and the Infimum in $\mathbb{R}$

Let $\mathbb{R}$ denote the real numbers, whose order $\le$ is that of an ordered field and in particular a total order, and whose addition and additive inverses are those of the underlying field; write $x-y$ for $x+(-y)$, and write $x<y$ to mean that $x\le y$ and $x\ne y$. Let $S\subseteq\mathbb{R}$ be nonempty, and let $\varepsilon\in\mathbb{R}$ satisfy $0<\varepsilon$.

If $S$ is bounded above, then $\sup S$ exists by the least upper bound property in The Real Numbers and Standard Notation and is unique by Uniqueness of the Supremum and of the Infimum. If $S$ is bounded below, then $\inf S$ exists and is unique by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below. In these two cases respectively the following hold.

1. (Strict form, above)¶ If $S$ is bounded above and $b\in\mathbb{R}$ satisfies $b<\sup S$, then there exists $s\in S$ with $b<s$.

2. (Strict form, below)¶ If $S$ is bounded below and $b\in\mathbb{R}$ satisfies $\inf S<b$, then there exists $s\in S$ with $s<b$.

3. (Epsilon form, above)¶ If $S$ is bounded above, then there exists $s\in S$ with $\sup S-\varepsilon<s$.

4. (Epsilon form, below)¶ If $S$ is bounded below, then there exists $s\in S$ with $s<\inf S+\varepsilon$.