June 2021 - Infinity is Really Big

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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