Principles Of Mathematical Analysis by Walter Rudin

A concise, rigorous introduction to undergraduate analysis that builds the real and complex number systems and develops point-set topology, sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, uniform convergence and power series, and basic multivariable theory including the implicit and inverse function theorems; tightly structured proofs and challenging exercises cultivate a precise, proof-based understanding useful for further study in analysis.