Real Analysis

This introductory lecture from MIT's Real Analysis course establishes foundational concepts in set theory and introduces formal proof writing, with an emphasis on the principle of mathematical induction for proving properties across natural numbers.

This lecture introduces Georg Cantor's foundational theory of set cardinality, defining injections, surjections, and bijections to formally compare the "size" of infinite sets. It rigorously demonstrates the countability of the rational numbers.

This lecture extends Cantor's foundational work on cardinality, progressing to ordered sets and the definition of the real numbers. It rigorously demonstrates the incompleteness of the rational numbers via their failure to satisfy the least upper bound property.

This lecture introduces the foundational axiomatic properties of fields and ordered fields, culminating in the complete characterization of the real numbers via the completeness axiom, a cornerstone of real analysis.

This lecture introduces the Archimedean property for real numbers, demonstrates the density of the rationals within the reals, and rigorously defines the absolute value function and its derived properties.

This lecture proves the uncountability of the real numbers using Cantor's diagonalization argument, establishing that the irrational numbers significantly outnumber the rationals. It also introduces the triangle inequality, sequences, and limits of real numbers, foundational concepts in analysis.

This lecture introduces the formal definition of convergent sequences in real analysis, focusing on the properties of monotone sequences and the utility of subsequences in proving convergence or divergence.

This lecture explicates the Squeeze Theorem for convergent sequences and demonstrates its application to various algebraic operations, including quotients, absolute values, and square roots.

This lecture introduces limit inferiors and limit superiors, then employs these concepts to rigorously prove the Bolzano-Weierstrass theorem, demonstrating that every bounded sequence possesses a convergent subsequence.

The session defines Cauchy sequences and establishes the completeness of the real numbers, proving every Cauchy sequence converges. It then transitions to infinite series, starting with geometric series.

This lecture delves into the convergence of infinite series, defining absolute convergence and proving the comparison test. Properties of p-series are also established.

This session introduces and proves three crucial tests for determining the convergence of infinite series: the ratio test, the root test, and the alternating series test.

This lecture initiates the study of limits of functions, introducing concepts such as cluster points and one-sided limits, which are foundational for understanding continuity.

This lecture explores the fundamental connection between limits of functions and limits of sequences, establishing how continuity provides a crucial link for proving results in single-variable calculus.

This lecture demonstrates the continuity of sine and cosine functions and explores algebraic properties of continuous functions. It also defines the Dirichlet function, highlighting its extensive discontinuities.

This lecture rigorously proves two cornerstone theorems of calculus: the Min/Max Theorem (Extreme Value Theorem) and Bolzano's Intermediate Value Theorem, examining the necessity of their hypotheses.

This lecture concludes the study of continuous functions by introducing uniform continuity, demonstrating its equivalence to continuity on closed and bounded intervals. It then establishes the formal definition of the derivative.

This lecture addresses whether continuity implies differentiability, constructing Weierstrass's famous continuous but nowhere differentiable function using the oscillatory properties of sine and cosine.

This lecture provides rigorous proofs for fundamental differentiation rules, including the product, quotient, and chain rules. It also establishes Rolle's Theorem and the Mean Value Theorem, exploring their applications.

This lecture covers Taylor's Theorem, derived as a direct consequence of repeated applications of the Mean Value Theorem. It also initiates the study of the Riemann integral, defining partitions and Riemann sums.

This lecture continues the study of the Riemann integral, demonstrating the convergence of Riemann sums for continuous functions on bounded intervals. It also begins exploring the essential properties of the Riemann integral.

This lecture proves key properties of the Riemann integral, including the triangle inequality, integration by parts, and the change of variable formula. It also establishes the Fundamental Theorem of Calculus.