A First Course in Real Analysis (Undergraduate Texts in Mathematics) - Tapa dura

This book is primarily intended to support a first course in real analysis, normally taken after a year of elementary calculus. The author gives a leisurely and thorough development of the properties of the field of real numbers. The topology of the real line is worked out using convergent sequences as the fundamental concept. The highlight of the book is the Fundamental Theorem of Calculus, proved for continuous functions in the context of the Riemann integral. An experimental final chapter gives an elementary exposition of the Fundamental Theorem of Calculus for the Lebesgue integral. Each section of the book is followed by a list of exercises (in all, nearly 400), often supplied with hints and many of them easy enough to be used on a test.