Completeness of the real numbers
Nested intervals theorem

The nested interval theorem is another form of completeness. Let I = be a sequence of closed intervals, and suppose that these intervals are nested in the sense that

Moreover, assume that b − a → 0 as n → +∞. The nested interval theorem states that the intersection of all of the intervals I contains exactly one point.

The rational number line does not satisfy the nested interval theorem. For example, the sequence (whose terms are derived from the digits of pi in the suggested way)

is a nested sequence of closed intervals in the rational numbers whose intersection is empty. (In the real numbers, the intersection of these intervals contains the number pi.)

Nested intervals theorem shares the same logical status as Cauchy completeness in this spectrum of expressions of completeness. In other words, nested intervals theorem by itself is weaker than other forms of completeness, although taken together with Archimedean property, it is equivalent to the others.
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