June 14, 2021July 1, 2021

Existence of nth Roots of Positive Reals

The Completeness Axiom helps us prove the existence of nth roots of positive real numbers, but the proof is quite challenging.

Real Analysis Study Help for Baby Rudin, Part 1.6

The existence of nth roots, including the square root of two, is ultimately based on the continuity of power functions with positive integer exponents. — Ultimately, the continuity of $f(x)=x^{2}$ at $x=\sqrt{2}$ allows us to prove the existence of $\sqrt{2}$ . Later, the argument is based on the Intermediate Value Theorem. In this article, the argument is

f(x)=x^{2} — Ultimately, the continuity of $f(x)=x^{2}$ at $x=\sqrt{2}$ allows us to prove the existence of $\sqrt{2}$ . Later, the argument is based on the Intermediate Value Theorem. In this article, the argument is

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June 7, 2021June 14, 2021

The Archimedean Property

The Completeness Axiom is a sufficient condition to prove the Archimedean Property, but is it necessary?

Real Analysis Study Help for Baby Rudin, Part 1.5

A Visualization of the Archimedean Property in the case where $x=8$ and $y=70$ . The smallest positive integer value of $n$ that makes $nx>y$ is $n=9$ .

x=8 — A Visualization of the Archimedean Property in the case where $x=8$ and $y=70$ . The smallest positive integer value of $n$ that makes $nx>y$ is $n=9$ .

Does the sequence $\left(\frac{1}{n}\right)_{n=1}^{\infty}$ approach zero as $n\rightarrow \infty$ ? Sure! After all, given any number

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April 1, 2021May 7, 2021

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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February 10, 2021February 10, 2021

Least Upper Bound (Supremum) in an Ordered Set

Suprema and Infima (Sups and Infs) Lie at the Heart of Real Analysis. The Range of Cos(n) is an Interesting Example.

Study Help for Baby Rudin, Part 1.3

For an ordered set $S$ and $M\subseteq S$ which is bounded above, the supremum of $M$ , when it exists in $S$ , is also called the least upper bound of $M$ in $S$ .

Which property separates the

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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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