Section: 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 $nx > y.$

  1. Definition: Real Intervals
  2. Definition: Supremum, Least Upper Bound
  3. Definition: Maximum (Real Numbers)
  4. Definition: Extended Real Numbers
  5. Definition: Supremum of Extended Real Numbers
  6. Definition: Infimum, Greatest Lower Bound
  7. Definition: Minimum (Real Numbers)
  8. Definition: Infimum of Extended Real Numbers
  9. Proposition: Closed Formula for the Maximum and Minimum of Two Numbers
  10. Definition: Nested Real Intervals
  11. Proposition: Limit of Nested Real Intervals

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