SC403201 MATHEMATICAL ANALYSIS I

1 The real numbers

        1.1 Natural numbers and induction

        1.2 Ordered fields

        1.3 The completeness axiom

        1.4 Topology of the real numbers

        1.5 Compact sets

2 Sequences

         2.1 Convergence

         2.2 Limit theorems

         2.3 Monotone sequences and Cauchy sequences

         2.4 Subsequences

3 Limits and Continuity

         3.1 Limit of functions

         3.2 Continuous functions

         3.3 Properties of continuous functions

         3.4 Uniformly continuity

4 Differentiation

         4.1 The derivative

         4.2 The mean value theorem

         4.3 L’Hospital’s rule

         4.4 Taylor’s theorem (optional)

5 Integration

          5.1 The Riemann integral

          5.2 Properties of the Riemann integral

          5.3 The fundamental theorem of calculus

6 Infinite series

          6.1 Convergence of infinite series

          6.2 Convergence tests

          6.3 Power series