What is the supremum of a set?

The least upper bound: an upper bound no greater than any other upper bound. It need not belong to the set.

What does Cauchy completeness mean for the real numbers?

Every Cauchy sequence of real numbers converges to a real number.

What does Dedekind completeness mean?

Every Dedekind cut in the ordered system is generated by a number in that system.

What does the nested interval theorem state?

Nested nonempty closed real intervals with lengths tending to zero have exactly one common point.

Why is the rational sequence of decimal approximations to $\pi$ not convergent in $\mathbb{Q}$?

It is Cauchy, but its limit $\pi$ is irrational and therefore is not in $\mathbb{Q}$.

Real analysisLesson 2 of 4 in the course Real numbers and sequences

Order, Sequence, and Interval Forms of Real-Number Completeness

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Completeness explains why the real number line has no gaps. In this lesson, we focus on two order-based ways to express this idea: the least-upper-bound property and Dedekind completeness.

A number $u$ is an upper bound of a set $S$ if every element of $S$ is less than or equal to $u$ . A least upper bound, also called a supremum, is an upper bound that is no larger than any other upper bound.

All flashcards

What is the supremum of a set?: The least upper bound: an upper bound no greater than any other upper bound. It need not belong to the set.
What does Cauchy completeness mean for the real numbers?: Every Cauchy sequence of real numbers converges to a real number.
What does Dedekind completeness mean?: Every Dedekind cut in the ordered system is generated by a number in that system.
What does the nested interval theorem state?: Nested nonempty closed real intervals with lengths tending to zero have exactly one common point.
Why is the rational sequence of decimal approximations to $\pi$ not convergent in $\mathbb{Q}$ ?: It is Cauchy, but its limit $\pi$ is irrational and therefore is not in $\mathbb{Q}$ .

Quiz questions

What does Cauchy completeness guarantee for a Cauchy sequence of real numbers?Answer: It converges to a real number.
Which conditions ensure that nested closed real intervals have exactly one common point?Answer: They are nonempty, each contains the next, and their lengths tend to zero.
What does the monotone convergence theorem say about a nondecreasing real sequence bounded above?Answer: It converges to the supremum of its terms.
What does the Bolzano–Weierstrass theorem guarantee for a bounded real sequence?Answer: It has a convergent subsequence.
In an ordered field, what additional property helps make Cauchy completeness equivalent to stronger standard forms of completeness?Answer: The Archimedean property.

Based on the Wikipedia article Completeness of the real numbers, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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