General description
A first course in Mathematical Analysis aims to tackle some of the issues which are glossed over in the development of calculus. The central concepts of limit and continuity will be introduced and used to prove rigorously some of the fundamental theorems in analysis. These ideas play a basic part in the subsequent development of mathematics.

Aims
This module aims to tackle some of the issues which are glossed over in the development of the calculus. The central concepts of limit and continuity will be introduced and used to prove rigorously some of the fundamental theorems in analysis. The theoretical aspects of the subject will be developed in conjunction with the techniques required to solve problems.

Learning outcomes
On completion of this module, a student should be able to:

• determine solution sets of elementary inequalities;
• determine whether or not a set of real numbers is bounded;
• determine the supremum and infimum of bounded sets;
• describe the notion of a sequence of real numbers and determine whether sequences are convergent or divergent;
• apply the standard theorems on convergence of sequences;
• manipulate sequences defined by recurrence relationships;
• use using the basic tests for convergence of series;
• compute the higher derivatives of certain functions and use Leibnitz's theorem;
• state and use the mean-value theorem of the differential calculus, Taylor's theorem and Maclaurin's theorem;
• establish inequalities by the use of differential calculus;
• evaluate simple integrals by consideration of Riemann sums;
• evaluate integrals with reduction formulae.

Syllabus
1. INEQUALITIES: Solution sets for rational inequalities.
2. BOUNDED SETS: Upper bound, lower bound, infimum, supremum. Completeness axiom for the real numbers.
3. SEQUENCES: Limit of a convergent sequence of real numbers. Formal derivation of some limit theorems. The sandwich theorem. Sequences defined by recurrence relationships. Increasing and decreasing sequences and related convergence theorems. Boundedness of convergent sequences. Subsequences.
4. APPLICATIONS OF THE DIFFERENTIAL CALCULUS: Rolle's theorem. Mean-value theorem of the differential calculus. Establishing inequalities using differential calculus. L'Hopital's rule. Leibnitz's theorem. Taylor's theorem, Maclaurin's theorem.
5. INFINITE SERIES: Partial sums. Convergence of infinite series. Examples of convergent and divergent series, including geometric series. Tests for convergence of series of positive terms: comparison test, ratio test, integral test.
6. INTEGRATION: Riemann sums. The definite integral as a limit of Riemann sums. Mean-value theorems for integrals. Fundamental theorem of the integral calculus. Reduction formulae.