An Introduction to Mathematical Analysis
Most of this book comprises a standard introduction to real analysis, from the number system and its completeness, through the elementary topology of the real line, sequences, limits and the differential calculus, the Riemann integral via step functions, series and uniform convergence. There are appendices dealing with countability and sets of measure zero.
The feature which distinguishes this book is the careful attention given to proof and its underlying logic. A chapter "An introduction to mathematical grammar'' gives a thorough discussion of the logical language used in mathematics and how it is used to construct proofs. This approach relates much more closely to the rest of a student's mathematical experience than does a discussion of truth tables and propositional calculus which some books contain.
Lewin, J., & Lewin, M. (1988). An introduction to mathematical analysis. New York: Random House.