Real Analysis builds the theory behind the calculus directly from the basic concepts of real numbers, limits and open and closed sets of in Rn. It gives the three characterizations of continuity: via epsilon-delta, sequences, and open sets. It gives three characterizations of compactness: as “closed and bounded,” via sequences, and via open covers. Topics include Fourier series, the Gamma function, metric spaces, and Ascoli’s Theorem. The text not only provides efficient proofs, but also shows students how to come up with them. The excellent exercises come with select solutions in the back. Here is a real analysis text that is short enough for the student to read and understand and complete enough to be the primary text for a serious undergraduate course.

• Numbers and logic
• Infinity
• Sequences
• Functions and limits

• Open and closed sets
• Continuous functions
• Composition of functions
• Subsequences
• Compactness
• Existence of maximum
• Uniform continuity
• Connected sets and the intermediate value theorem
• The Cantor set and fractals

• The derivative and the mean value theorem
• The Riemann integral
• The fundamental theorem of calculus
• Sequences of functions
• The Lebesgue theory
• Infinite series ? a to n
• Absolute convergence
• Power series
• Fourier series
• Strings and springs
• Convergence of Fourier series
• The exponential function
• Volumes of n-balls and the gamma function

• Metric spaces
• Analysis on metric spaces
• Compactness in metric spaces
• Ascoli''s theorem
• Partial solutions to exercises
• Greek letters