The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.
- Prelude to Real Analysis
- 1: Numbers - Real (ℝ) and Rational (ℚ)
- 2: Calculus in the 17th and 18th Centuries
- 3: Questions Concerning Power Series
- 4: Convergence of Sequences and Series
- 5: Convergence of the Taylor Series: A “Tayl” of Three Remainders
- 6: Continuity - What It Isn’t and What It Is
- 7: Intermediate and Extreme Values
- 8: Back to Power Series
- 9: Back to the Real Numbers
- Epilogue to Real Analysis
Thumbnail: Real number line with some constants such as \(\pi\). Image used with permission (Public Domain; User:Phrood).
Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)