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searching for Completeness of the real numbers 5 found (17 total)

alternate case: completeness of the real numbers

Cauchy sequence (3,225 words) [view diff] exact match in snippet view article find links to article

Bolzano–Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and

Complete metric space (2,522 words) [view diff] exact match in snippet view article find links to article

however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different

Extreme value theorem (3,937 words) [view diff] exact match in snippet view article find links to article

boundedness theorem, f is bounded from above, hence, by the Dedekind-completeness of the real numbers, the least upper bound (supremum) M of f exists. It is necessary

Reverse mathematics (4,781 words) [view diff] exact match in snippet view article find links to article

assertions are equivalent to ACA0 over RCA0: The sequential completeness of the real numbers (every bounded increasing sequence of real numbers has a limit)

Nested intervals (4,102 words) [view diff] exact match in snippet view article find links to article

Königsberger, Konrad (2003), "2.3 Die Vollständigkeit von R (the completeness of the real numbers)", Analysis 1, 6. Auflage (6th edition), Springer-Lehrbuch