Real Analysis

Calendar

LEC #	TOPICS	KEY DATES
1	Sets, Set Operations, and Mathematical Induction
2	Cantor’s Theory of Cardinality (Size)	Assignment 1 due
3	Cantor’s Remarkable Theorem and the Rationals’ Lack of the Least Upper Bound Property
4	The Characterization of the Real Numbers	Assignment 2 due
5	The Archimedian Property, Density of the Rationals, and Absolute Value
6	The Uncountabality of the Real Numbers	Assignment 3 due
7	Convergent Sequences of Real Numbers
8	The Squeeze Theorem and Operations Involving Convergent Sequences	Assignment 4 due
9	Limsup, Liminf, and the Bolzano-Weierstrass Theorem
10	The Completeness of the Real Numbers and Basic Properties of Infinite Series	Assignment 5 due
11	Absolute Convergence and the Comparison Test for Series
12	The Ratio, Root, and Alternating Series Tests	Assignment 6 due
	Midterm Exam	Midterm Exam due
13	Limits of Functions
14	Limits of Functions in Terms of Sequences and Continuity	Assignment 7 due
15	The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet’s Function
16	The Min/Max Theorem and Bolzano’s Intermediate Value Theorem	Assignment 8 due
17	Uniform Continuity and the Definition of the Derivative
18	Weierstrass’s Example of a Continuous and Nowhere Differentiable Function	Assignment 9 due
19	Differentiation Rules, Rolle’s Theorem, and the Mean Value Theorem
20	Taylor’s Theorem and the Definition of Riemann Sums	Assignment 10 due
21	The Riemann Integral of a Continuous Function
22	The Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula	Assignment 11 due
23	Pointwise and Uniform Convergence of Sequences of Functions
24	Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits	Assignment 12 due
25	Power Series and the Weierstrass Approximation Theorem
	Final Assignment	Final Assignment due