Introduction to Mathematical Analysis

Description

Subject objectives: Introduce students, with essential support from examples and practice, to the understanding of the first structure of Mathematical Analysis, the ordered and complete field of real numbers, and the fundamentals of real-valued real functions. Introduce and consolidate, through examples and exercises, the notions of convergence of sequences and numerical series. Present, with practice using different notations, operations with complex numbers. Introduce the different notions of limits of real-valued real functions, and study continuity and uniform continuity of these functions. Contents: 1. REAL NUMBERS (approx. 8 lecture classes) 1.1 Natural numbers. Principle of induction. 1.2 Rational numbers. Countability. 1.3 Axiomatic structure of the real numbers (R). Supremum axiom and consequences. 1.4 Archimedean property of R. Density of Q in R. 2. SEQUENCES OF REAL NUMBERS (approx. 9 lecture classes) 2.1 Intuitive introduction to the concepts of sequence and limit. General notions. 2.2 Convergent sequences and their limits. Properties. 2.3 Infinite limits. 2.4 Convergence and divergence of monotonic sequences. 2.5 Subsequences. Bolzano-Weierstrass Theorem. Oscillation limits. 2.6 Cauchy sequences. Completeness of ℝ. 2.7 Limit calculations. Stirling’s and Stolz’s criteria. 3. SERIES OF REAL NUMBERS (approx. 8 lecture classes) 3.1 Intuitive introduction to the concept of series and its sum. 3.2 Numerical series. Convergence of series. 3.3 Series with non-negative terms. Convergence criteria. 3.4 Absolute and conditional convergence. Non-absolute convergence criteria. 3.5 Decimal expression in R and other numeral systems. 4. COMPLEX NUMBERS (approx. 2 lecture classes) 4.1 Complex numbers. Binomial form and basic operations. 4.2 Exponential form and its consequences: powers, roots, Euler’s and De Moivre’s formulas. 5. LIMITS (approx. 7 lecture classes) 5.1 Topological preliminaries in R. 5.2 Limit of a function at a point. 5.3 Lateral limits. 5.4 Infinite limits and limits at infinity. 5.5 Calculating limits: Indeterminate forms. 6. CONTINUITY (approx. 8 lecture classes) 6.1 Continuity of a function at a point. 6.2 Sequential continuity. 6.3 Continuous functions: Properties. 6.4 Weierstrass and Bolzano theorems. 6.5 Continuity of monotonic functions and their inverses. 6.6 Uniform continuity. 6.7 Heine’s Theorem. 6.8 Continuous extension theorem. 6.9 Sufficient and necessary criteria for uniform continuity.