Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $x,y\in\mathbb{R}$.

¶ If $x\le y+\varepsilon$ for every real $\varepsilon>0$, then $x\le y$.

¶ If $y-\varepsilon\le x$ for every real $\varepsilon>0$, then $y\le x$.

¶ If $0\le x$ and $x\le\varepsilon$ for every real $\varepsilon>0$, then $x=0$.

Let now $S$ be a nonempty subset of $\mathbb{R}$, with bounds, suprema and infima as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order.

¶ If $S$ is bounded above, then for every $b\in\mathbb{R}$ with $b<\sup S$ there is $s\in S$ with $b<s$.

¶ If $S$ is bounded above, then for every real $\varepsilon>0$ there is $s\in S$ with $\sup S-\varepsilon<s$.

¶ If $S$ is bounded below, then for every $b\in\mathbb{R}$ with $\inf S<b$ there is $s\in S$ with $s<b$.

¶ If $S$ is bounded below, then for every real $\varepsilon>0$ there is $s\in S$ with $s<\inf S+\varepsilon$.