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 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
Does the sequence approach zero as ? Sure! After all, given any number
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
The Completeness Axiom for the real number system is intimately tied to the concept of the supremum of a set of
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
Which property separates the
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
The set of real numbers is more than just a set. As
Baby Rudin: Let Me Help You Understand It!
The First in a Series of Blog Posts on Baby Rudin
Study Help for Baby Rudin, Part 1.1
The textbook “Principles of Mathematical Analysis”, by Walter Rudin, is considered a “classic” text on real analysis, the subject that uses rigorous deductive
The Fundamental Theorem of Calculus
Calculus 2, Lectures 5A through 6 (Videotaped Fall 2016)
The Fundamental Theorem of Calculus is often split into two forms in textbooks.
These forms are typically called the “First Fundamental Theorem of Calculus” and the “Second Fundamental Theorem of … Read the rest
Integration by Parts (and Linear Motion)
Calculus 2, Lectures 3B through 4 (Videotaped Fall 2016)
You might say that integration by parts is an “unexpected” or “surprising” method of integration.
While the other main method, integration by substitution, can be thought … Read the rest
Integration by Substitution (Method of Integration)
Calculus 2, Lectures 2A through 3A (Videotaped Fall 2016)
In Calculus 1, the techniques of integration introduced are usually pretty straightforward. In fact, they are usually just memorized as basic facts about antiderivatives.
For Calculus 2, various new integration techniques are introduced, including integration by … Read the rest