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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:

1. either $$a$$ is “equal” $$b$$, or
2. $$a$$ is “smaller” than $$b$$, or
3. $$a$$ 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 $nx > y.$

| | | | created: 2014-02-20 22:17:01 | modified: 2020-07-06 11:55:56 | by: bookofproofs