實數集合 The set of real numbers Field axioms; order axioms; completeness axiom; supremum and infimum.

Chapter 2

數列的收斂性 Convergent sequences Limit of a sequence; properties of limits; monotone convergence theorem, Bolzano-Weierstrass theorem, nested interval theorem and Cauchy sequences; subsequences

Chapter 3

函數的極限與連續性 Limits and continuity of functions Limit of a function and its properties; continuity of a function and its properties; extreme value theorem; intermediate value theorem; uniform continuity; examples of continuous functions; Rolle’s theorem and mean value theorem.

Chapter 4

黎曼積分理論 Riemann integrals Upper sum and lower sum; integrable functions; integrability condition; continuous and monotone functions; Riemann sum; Darboux theorem, Fundamental Theorem of Calculus; properties of integrals; inequalities.

Chapter 5

泰勒展開式定理和冪級數 Taylor expansion theorem and power series Taylor expansion theorem; Taylor series and Maclaurin series; uniform convergence; power series and its properties.

References:

W.R. Wade, An Introduction to Analysis, fourth edition, Pearson, 2010. ISBN 0-13-615370-4

P.M. Fitzpatrick, Advanced Calculus, second edition, Thomson Brook/Cole, 2006. ISBN 0-534-37603-7.