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In summary, the Bolzano-Weierstrass theorem is a fundamental theorem in real analysis that states that any bounded sequence of real numbers has a convergent subsequence. It is related to the Axiom of Completeness, which guarantees the existence of a least upper bound for any non-empty set of real numbers that is bounded above. The Axiom of Completeness is used to prove the Bolzano-Weierstrass theorem by showing that a bounded sequence can be divided into two subsequences, one bounded above and one bounded below. Another way to prove the theorem is by using the nested interval theorem, which states that the intersection of nested, decreasing intervals is non-empty.

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Now what x would be in your set S? Can you see any x ≤ 0 is in S but any x > 0 isn't?

The Bolzano-Weierstrass theorem, also known as the Weierstrass theorem, is a fundamental theorem in real analysis that states that any bounded sequence of real numbers has a convergent subsequence.

The Axiom of Completeness, also known as the Completeness Axiom or the least upper bound property, is one of the axioms of the real number system. It states that every non-empty set of real numbers that is bounded above has a least upper bound.

The Axiom of Completeness is used to prove the Bolzano-Weierstrass theorem. It provides a way to show that a bounded sequence of real numbers has a convergent subsequence by guaranteeing the existence of a least upper bound.

The proof involves showing that a bounded sequence of real numbers can be divided into two subsequences, one that is bounded above and one that is bounded below. Using the Axiom of Completeness, it can be shown that the bounded above subsequence has a least upper bound, which is the limit of the original sequence.

Yes, there are other ways to prove the Bolzano-Weierstrass theorem. One common method is to use the nested interval theorem, which states that if a sequence of closed intervals is nested and has a decreasing length, then the intersection of all the intervals is non-empty. This can be used to show the existence of a convergent subsequence in a bounded sequence of real numbers.

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