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Real Intervals and Bounded Real Sets

When we introduced the number system of real numbers \(\mathbb R\), we saw that this field as an ordered field. This means that any given two real numbers \(a,b\in \mathbb R\) can be compared with each other. This comparison always has exactly one of the following results:

either \(a\) is “equal” \(b\), or

\(a\) is “smaller” than \(b\), or

\(a\) is “greater” than \(b\).

Moreover, it is Archimedean, meaning that the ordering is “regular” in the sense that for any two positive real numbers $x,y > 0$, there exist a natural number $n$ such that $nx > y.$