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

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 $n=9$ .*

*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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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

But what is a … Read the rest

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 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

Does the sequence approach zero as ? Sure! After all, given any number

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

Which property separates the

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.

Rational Numbers

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