Properties of the Supremum

The supremum and infimum of a bounded set of real numbers have many interesting properties

Study Help for Baby Rudin, Part 1.4

One of the properties of the supremum is that it is additive as a set function on sets of real numbers which are bounded above. — The supremum is additive as a set function on sets of real numbers which are bounded above.

The Completeness Axiom for the real number system is intimately tied to the concept of the supremum of a set of

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January 19, 2021June 3, 2021

Definitions of Ordered Set and Ordered Field

And an Ordered Field That Can Be Ordered in More Than One Way

Study Help for Baby Rudin, Part 1.2

A visual for a totally ordered set with three elements. — A graphic representation of the totally ordered set of three points $\left\{a,b,c\right\}$ with $a<b<c$ . The directed line segments are not part of the set. They only indicate the order.

A graphic representation of the totally ordered set of three points $\left\{a,b,c\right\}$ with $a<b<c$ . The directed line segments are not part of the set. They only indicate the order.

The set of real numbers $\Bbb{R}$ is more than just a set. As

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January 28, 2019July 26, 2019

Does the Square Root of Two Exist?

The proof that the square root of two is an irrational number is considered to be one of the most elegant in all of mathematics.

As a formal “If…, then…” statement, it reads: If $x^{2}=2,$ then $x$ is not a rational number. A more informal statement of this is “the square root of two is irrational”.

Rational Numbers

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