Limits

I Foundations.- 1 Sets and Functions.- 1.1 Sets.- 1.2 Ordered pairs.- 1.3 Functions.- 2 Real and Complex Numbers.- 2.1 Algebraic properties of real numbers.- 2.2 Order.- 2.3 Upper and lower bounds.- 2.4 Complex numbers.- 2.5 Notation.- II Limits.- 3 Limits.- 3.1 Introduction.- 3.2 Directed sets.- 3.3 The definition of a limit.- 3.4 Examples of limits.- 3.5 Sums, products, and quotients of limits.- 3.6 Limits and inequalities.- 3.7 Functions tending to infinity.- 4 Bisection Arguments.- 4.1 Nested intervals.- 4.2 The Intermediate Value Therem.- 4.3 The Mean Value Inequality.- 4.4 The Cauchy Criterion.- 5 Infinite Series.- 5.1 Infinite series.- 5.2 Unordered sums.- 5.3 Absolute convergence and rearrangements.- 5.4 The Cauchy Product.- 5.5 Iterated sums.- 6 Periodic Functions.- 6.1 The exponential function.- 6.2 The trigonometric functions.- 6.3 Periodicity and ?.- 6.4 The argument of a complex number.- 6.5 The logarithm.- III Analysis.- 7 Sequences.- 7.1 Convergent sequences.- 7.2 Some important examples.- 7.3 Bounded sequences.- 7.4 The Fundamental Theorem of Algebra.- 7.5 Unbounded sequences.- 7.6 Upper and lower limits.- 8 Continuous Functions.- 8.1 Continuous functions.- 8.2 Functions continuous on an interval.- 8.3 Monotonic functions.- 8.4 Uniform continuity.- 8.5 Uniform convergence.- 9 Derivatives.- 9.1 The derivative.- 9.2 The Chain Rule.- 9.3 The Mean Value Theorem.- 9.4 Inverse functions.- 9.5 Power series.- 9.6 Taylor series.- 10 Integration.- 10.1 The integral.- 10.2 Upper and lower integrals.- 10.3 Integrable functions.- 10.4 Integration and differentiation.- 10.5 Improper integrals.- 10.6 Integration and differentiation of series.- 11 ?, ?, e, and n!.- 11.1 The number e.- 11.2 The number ?.- 11.3 Euler’s constant ?.- 11.4 Stirling’s formula for n!.- 11.5 A series and an integral for ?.- Appendix: Mathematical Induction.- References.