Mathematical Analysis Course

Lesson 1 • Proof Techniques and Strategies

Introduces direct proof, contradiction, contrapositive, and induction. Builds the toolkit students apply when verifying every analytic result.

Lesson 2 • Axiomatic Structure of Real Numbers

Presents field axioms, order axioms, and the completeness axiom. These axioms underpin every theorem about limits, continuity, and integration.

Lesson 3 • Functions and Cardinality

Defines injections, surjections, and bijections, then classifies infinite sets. Cardinality concepts reappear when studying countable unions and measure.

Lesson 1 • Monotone Sequences and Subsequences

Establishes the monotone convergence theorem and Bolzano-Weierstrass theorem. These results are foundational for series convergence and compactness arguments.

Lesson 2 • Limit Laws and Algebraic Operations

Proves sum, product, and quotient rules for sequence limits. Enables efficient computation of limits without repeating epsilon-N arguments each time.

Lesson 3 • Convergence and the Epsilon-N Definition

Formalizes what it means for a sequence to converge using epsilon-N. Connects intuitive notions of closeness to the completeness axiom introduced earlier.

Lesson 1 • Continuity and Types of Discontinuity

Defines pointwise and uniform continuity and classifies removable, jump, and essential discontinuities. Prepares students for integration theory requiring continuity conditions.

Lesson 2 • Key Theorems on Continuous Functions

Proves the intermediate value theorem and extreme value theorem. These results justify root-finding algorithms and optimisation methods used in applied chapters.

Lesson 3 • Uniform Continuity and Extensions

Deepens uniform continuity with the Heine-Cantor theorem and extension results. Connects to integration by showing uniformly continuous functions are Riemann integrable.

Lesson 1 • Taylor's Theorem and Approximation

Derives Taylor's theorem with Lagrange and Peano remainder forms. Students use polynomial approximations to estimate function values and analyse local behaviour.

Lesson 2 • Differentiation Rules and Higher Derivatives

Proves sum, product, quotient, and chain rules rigorously. Higher-order derivatives are introduced in preparation for Taylor series expansions.

Lesson 3 • Mean Value Theorems and Applications

Proves Rolle's theorem, the mean value theorem, and Cauchy's generalised form. These theorems underlie monotonicity tests, error bounds, and L'Hôpital's rule.

Lesson 4 • L'Hôpital's Rule and Indeterminate Forms

Proves L'Hôpital's rule using Cauchy's mean value theorem and applies it to all indeterminate forms. Reinforces the interplay between limits and derivatives.

Lesson 1 • Properties of the Riemann Integral

Establishes linearity, monotonicity, and additivity of the integral over subintervals. These properties are used repeatedly in series and improper integral chapters.

Lesson 2 • Classes of Integrable Functions

Proves that continuous and monotone functions are Riemann integrable. Identifies functions with finitely many discontinuities as integrable, expanding the applicable class.

Lesson 3 • Improper Integrals and Convergence

Extends the Riemann integral to unbounded intervals and unbounded integrands. Convergence tests for improper integrals prepare students for series and Fourier analysis.

Lesson 4 • Partitions, Upper and Lower Sums

Defines partitions, Darboux sums, and the Riemann integral through the sup-inf construction. Grounds integration in the completeness of the reals established earlier.

Lesson 1 • Series, Partial Sums, and Basic Tests

Defines series convergence via partial sums and proves the divergence test and geometric series formula. Connects series to sequence convergence studied in Chapter 2.

Lesson 2 • Absolute and Conditional Convergence

Distinguishes absolute from conditional convergence and proves the Riemann rearrangement theorem. Highlights the structural difference between the two convergence types.

Lesson 3 • Ratio, Root, and Alternating Series Tests

Establishes ratio and root tests using limsup and proves the alternating series test. Students learn to select the most efficient test for a given series.

Lesson 4 • Power Series and Radius of Convergence

Defines power series, computes radii of convergence, and proves term-by-term differentiation and integration. Lays the groundwork for the analytic functions chapter.

Lesson 1 • Preservation Theorems Under Uniform Convergence

Proves that uniform limits of continuous functions are continuous and that limits and integrals interchange. These theorems justify term-by-term operations on series.

Lesson 2 • Pointwise and Uniform Convergence

Defines pointwise and uniform convergence and contrasts them with examples. Uniform convergence is identified as the condition needed to interchange limits and operations.

Lesson 3 • Weierstrass Approximation Theorem

States and proves the Weierstrass approximation theorem using Bernstein polynomials. Demonstrates that continuous functions on closed intervals are uniformly approximable by polynomials.

Lesson 4 • Taylor and Analytic Functions

Defines analytic functions as those equal to their Taylor series and identifies conditions for Taylor series validity. Extends differentiation theory to infinitely differentiable functions.

Lesson 1 • Compactness in Metric Spaces

Characterises compactness via sequential compactness and total boundedness. Proves the Heine-Borel theorem in Euclidean space as a special case.

Lesson 2 • Connectedness and Contraction Mappings

Defines connected and path-connected metric spaces and proves the Banach fixed-point theorem. The contraction mapping principle has direct applications in differential equations.

Lesson 3 • Metric Spaces: Definitions and Examples

Introduces the metric space axioms and surveys key examples including Euclidean, discrete, and function spaces. Provides the abstract framework that subsumes real-line analysis.

Lesson 4 • Convergence and Completeness in Metric Spaces

Extends Cauchy sequences and completeness to metric spaces and proves the Baire category theorem. Completeness is shown to be a metric, not topological, property.