Descrição do livro
Calculus is usually taught as a toolkit: rules for derivatives, tricks for integrals, formulas for series. It works — but why does any of it actually work?
Real Analysis is a guided introduction to the ideas that answer that question. Starting from the completeness of the real numbers, it builds up sequences, series, continuity, differentiation, and Riemann integration on precise, rigorous foundations — the same foundations professional mathematicians rely on.
Every definition is paired with the intuition behind it. Every theorem comes with a full proof and commentary on the idea driving that proof, so arguments feel discovered rather than memorized. Along the way you'll meet some of the most surprising results in undergraduate mathematics: a function continuous everywhere but differentiable nowhere, a convergent series that can be rearranged to sum to any number you like, and a sequence of continuous functions whose limit isn't continuous at all.
Topics covered:
- The real number system and completeness
- Sequences, limits, and series
- Topology of the real line and compactness
- Continuity and uniform continuity
- Differentiation and the Mean Value Theorem
- Riemann integration and the Fundamental Theorem of Calculus
- Uniform convergence, power series, and Taylor series
- An introduction to metric spaces
- A dedicated chapter on how to read and write analysis proofs
Written for students who already know calculus and want to understand why it's true — no prior proof-writing experience required.
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