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In any metric space, a Cauchy sequence is bounded (since for some N, all terms of the sequence from the N-th onwards are within distance 1 of each other, and if M is the largest distance between and any terms up to the N-th, then no term of the sequence has distance greater than + from ).
Because () is bounded, this sequence has a lower bound and an upper bound . We take I 1 = [ s , S ] {\displaystyle I_{1}=[s,S]} as the first interval for the sequence of nested intervals. Then we split I 1 {\displaystyle I_{1}} at the mid into two equally sized subintervals.
It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges. Let S be a nonempty set of real numbers. If S has exactly one element, then its only element is a least upper bound. So consider S with more than one element, and suppose that S has an upper bound B 1.
Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed and bounded interval is uniformly continuous. Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof. [1]
A metric space is said to be totally bounded if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and totally bounded. [2] Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded).
After formally defining the convergence of sequences and accumulation points of sequences, one can also prove the Bolzano–Weierstrass theorem using nested intervals. In a follow-up, the fact, that Cauchy sequences are convergent (and that all convergent sequences are Cauchy sequences) can be proven. This in turn allows for a proof of the ...
The monotone convergence theorem (described as the fundamental axiom of analysis by Körner [1]) states that every nondecreasing, bounded sequence of real numbers converges. This can be viewed as a special case of the least upper bound property, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers.
It states that a decreasing nested sequence () of non-empty, closed and bounded subsets of has a non-empty intersection. This version follows from the general topological statement in light of the Heine–Borel theorem , which states that sets of real numbers are compact if and only if they are closed and bounded.