Mathematical AnalysisIt provides a transition from elementary calculus to advanced courses in real and complex function theory and introduces the reader to some of the abstract thinking that pervades modern analysis. 
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A classic introduction that is just that.
Review: Mathematical Analysis
User Review  GoodreadsProbably the only mathematical text I have ever read covertocover. A very good introduction to Analysis as long as you already have some maths background. Naturally as a pure maths text it is very dry, with some positively arid sections, even for one interested in learning the subject. Read full review
Contents
The Real and Complex Number Systems  1 
Some Basic Notions of Set Theory  32 
Elements of Point Set Topology  47 
Copyright  
15 other sections not shown
Common terms and phrases
absolutely convergent accumulation point analytic apply Theorem assume that f axioms bounded variation called Cauchy Cauchy sequence CauchyRiemann equations compact interval complex numbers complexvalued function constant contains continuous functions converges absolutely countable collection curve Definition denote differentiable disjoint disk dx exists equation example Exercise f is continuous finite number formula Fourier series function defined function f given Hence implies inequality infinite integer interior point Jacobian Lebesgue integral Let f limit function linear matrix MeanValue Theorem measure metric space nball nonempty nonnegative obtain onedimensional onetoone open interval open set partial derivatives partial sums partition positive integers power series properties prove rational numbers real numbers realvalued function Riemann integral RiemannStieltjes integral satisfies step functions subinterval theorem shows transformation uniform convergence upper functions variable vector vectorvalued function write zero