Understanding AnalysisThis book outlines an elementary, onesemester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. 
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LibraryThing Review
User Review  aethercowboy  LibraryThingIf you have ever wanted to push the envelope for your mathematical learning, I suggest reading this book. It is an overview of mathematical analysis, involving rigorous proofs. The book is small, but it is filled to the brim with knowledge. A must for any serious student of mathematics. Read full review
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This book has a very clear writing style, and the exercises are illustrative of the concepts discussed. I also like that the author grounds the math in a historical context it makes the material richer for the reader.
I have looked at a lot of analysis books, and this is among the best. Another I really like is The Way of Analysis. I think these two are some of the best for understanding the material.
Contents
II  xi 
IV  2 
V  11 
VI  16 
VII  27 
VIII  31 
IX  33 
XI  36 
XXXIII  125 
XXXIV  127 
XXXVII  129 
XXXVIII  135 
XXXIX  142 
XL  146 
XLI  149 
XLII  152 
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Algebraic Limit Theorem analysis arbitrary argue argument assume Axiom of Completeness Cantor set Cauchy Criterion Cauchy sequence Chapter closed interval closed sets compact sets compute conclude construct contains continuous functions convergent sequence converges uniformly Definition derivative differentiable function Dirichlet's function discontinuities discussion domain element endpoints equation example Exercises Exercise f is continuous f is integrable f uniformly fact finite number fn(x follows function f functional limits Given implies infinite series intersection irrational numbers least upper bound Let f limit point lower sums mathematical Mean Value Theorem measure zero metric space natural numbers neighborhood Nested Interval nonempty open interval open sets partial sums pointwise polynomials power series prove rational numbers real numbers result Riemann integral rigorous satisfying Section sequence xn series converges set A C R Show sin(nx statement subinterval subset tagged partition Taylor series uncountable uniform convergence uniformly continuous Weierstrass