Comparison of Real Numbers with Arbitrary Positive Slack

Let $\mathbb{R}$ be the ordered field of real numbers, with order $\le$, and let $a,b\in\mathbb{R}$. Then the following hold.

1. (Slack above)(i) If $a\le b+\varepsilon$ for every positive $\varepsilon\in\mathbb{R}$, then $a\le b$.

2. (Slack below)(ii) If $b-\varepsilon\le a$ for every positive $\varepsilon\in\mathbb{R}$, then $b\le a$.

3. (Vanishing)(iii) If $0\le a$ and $a\le\varepsilon$ for every positive $\varepsilon\in\mathbb{R}$, then $a=0$.