KindiakMath

Real Analysis

Here are my write-ups on the theory of real analysis and some non-trivial exercises therein. Any serious study in logically rigorous mathematics will inevitably require the tools and techniques in real analysis. Enjoy!

Theory

Real Numbers

  1. A Rational Prelude
  2. The Problem with Rationals
  3. The Supremum Superman
  4. Constructing the Reals

Limits and Continuity

  1. The Foundation of Real Analysis
  2. The Beauty of Bounded Sequences
  3. Theoretical Continuity
  4. Generalising Continuity

Special Functions

  1. What is a Square Root?
  2. Defining the Rational Exponential
  3. Defining the Real Exponential
  4. What is the Exponential Unit?

Series

  1. Cauchy’s True Utility
  2. Acquainting with Convergent Series
  3. A Handful of Convergence Tests
  4. The Limit of Functions
  5. Adding Infinitely Many Functions

Integration

  1. Technical Integration
  2. The Integrable Limit Theorem
  3. Generalising Integration

Exercises

Real Numbers

  1. When Integers Converge
  2. Mashed Potato Rationals
  3. Kronecker Approximation
  4. Principle of Real Induction

Limits and Continuity

  1. The Limit of Quotients
  2. Investigating the Limit Superior
  3. Several Limit Comparisons
  4. A Lovely Approximation Theorem
  5. Several Continuity Exercises
  6. Functional Reverse-Engineering
  7. Stirling’s Approximation

Series

  1. Strengthening Point-wise Convergence
  2. Dini’s Mutated Result
  3. The Bernstein Approximants
  4. Verifying the Euler Method
  5. An Unnatural Logarithm
  6. Controlling Convergence

Integration

  1. Flavors of Integrability
  2. The Popcorn Function
  3. Composing Integrability
  4. Several Integrable Combinations
  5. Relaxing the Fundamental Theorem
  6. A Less-Dominated Convergence Theorem
  7. Generalising the Factorial