Elements of Mathematical Analysis

Many problems in mathematics cannot be solved explicitly. So one resorts to finding approximate solutions and estimate the error between a true solution and the approximate one. Indeed, one may even be able to demonstrate the existence of a solution by exhibiting a sequence of approximate solutions that converge to an exact solution. The study of limiting processes is the central theme in mathematical analysis. It involves the quantification of the notion of limit and precise formulation of intuitive notions of infinite sums, functions, continuity and the calculus.

This course is a rigorous analysis of the real numbers, as well as an introduction to writing and communicating mathematics well. Topics will include: construction of the real numbers, fields, complex numbers, topology of the reals, metric spaces, careful treatment of sequences and series, functions of real numbers, continuity, compactness, connectedness, differentiation, the mean value theorem, integration, with an introduction to sequences of functions.

Content:

1.  Convergence of a nonrecurring decimal and the completeness axiom in the form that an increasing sequence which is bounded above converges to a real number.
2. The completeness axiom as the main distinguishing feature between the rationals and the reals; approxiamtion of irrationals by rationals and vice-versa.
3.  Formal definition of sequence and subsequence.
4.  Limit of a sequence of real numbers; Cauchy sequences and the Cauchy criterion.
5. Series: (a) Series with positive terms, (b) Alternating series.
6.  Continuity, Continuous Limits.
7. Differentiability, Properties of differentiable functions.
8. Power Series.
9. Riemann Integral, Riemann-Stieltjes Integral
10. Sequence of functions

Objectives:
By the end of the module the student should be able to:

• Understand what is meant by the symbol 'infinity'
• Understand what it means for a sequence to converge or diverge and to compute simple limits
• Determine when it makes sense to add up infinitely many numbers
• Understand the notions of continuity and differentiability
• Establish various properties of continuous and differentiable functions
• Answer the question "when can a function be represented by a power series?"
• Develop their own methods for solving problems
• Learn the content of real analysis.
• Learn to read and write rigorous proofs.
• Learn good mathematical writing skills and style.

Textbook(s):

1. Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill. We will cover Chapters 1 through 7. There are also many other books on analysis that you may wish to consult in the library, for example
2. M. Spivak, Calculus, Benjamin.
3. M. Hart, Guide to Analysis, Macmillan. (A good traditional text with theory and many exercises.)