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Wyatt's Notes — University

Real Analysis

Real Analysis

Real analysis reconstructs the calculus on a rigorous foundation. Where the introductory calculus takes the real numbers, limits, and continuity as informal notions, real analysis defines each precisely and proves the theorems that justify the operations of differentiation and integration.

The development proceeds in the standard order: the completeness of the real number system is established first, because every subsequent result (the Monotone Convergence Theorem, the Bolzano-Weierstrass Theorem, the Extreme Value Theorem) depends on it. Sequences and series are then treated rigorously, followed by the topology of the real line, continuity, differentiability, and finally the Riemann integral. The section closes with the uniform convergence of sequences and series of functions, the condition under which limits commute with integration and differentiation.

Contents

  1. The Real Number System
  2. Sequences and Limits
  3. Series
  4. Continuity
  5. Differentiability
  6. Riemann Integration
  7. Sequences and Series of Functions
  8. Problem Set

Overview

University-level real analysis notes covering rigorous calculus, sequences, and integration.

Topics Covered

  • Real Number System: Completeness, Archimedean property, density of rationals
  • Sequences and Limits: Convergence, monotone convergence, Bolzano-Weierstrass
  • Series: Convergence tests, absolute convergence, power series
  • Continuity and Differentiability: Limits, derivatives, mean value theorem

Prerequisites

  • Calculus (sequences, series, differentiation, integration)
  • Basic logic and mathematical proofs
  • Mathematical maturity

How to Use These Notes

Start with the real number system to build foundational knowledge, then progress to sequences, series, and calculus. Each section includes worked examples and practice problems.

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Additional Resources

Each section includes:

  • Detailed explanations of key concepts
  • Worked examples with step-by-step solutions
  • Practice problems with answers
  • Common pitfalls and how to avoid them
  • Connections to other areas of mathematics

Study Tips

  1. Master the definitions: Real analysis requires precise understanding of limits and continuity
  2. Practise proofs: Learn to write clear, rigorous proofs
  3. Draw diagrams: Visualise sequences, series, and functions
  4. Learn standard examples: Know the properties of common functions (polynomials, trigonometric, exponential)
  5. Connect to calculus: Relate real analysis to the calculus you already know