Faculty evidence

Lesson Keys

Use keys to evaluate reasoning, not merely final statements. Accept equivalent valid proofs.

U01 L01

Ordered-Field Axioms and Consequences

U01 L02

Absolute Value, Intervals and the Triangle Inequality

U01 L03

Upper Bounds, Suprema and Infima

U01 L04

The Completeness Axiom

U01 L05

Archimedean Property and Density of the Rationals

U01 L06

Nested Intervals and Monotone Completeness

U01 L07

Proof Synthesis: Foundations of R

U02 L01

Sequence Limits from the Epsilon Definition

U02 L02

Limit Uniqueness and Algebra of Limits

U02 L03

Monotone Convergence Theorem

U02 L04

Subsequences and Cluster Points

U02 L05

Bolzano–Weierstrass Theorem

U02 L06

Cauchy Sequences and Completeness

U02 L07

Limit Superior, Limit Inferior and Oscillation

U03 L01

Series, Partial Sums and the Cauchy Criterion

U03 L02

Positive-Term Comparison Tests

U03 L03

Ratio, Root and Integral Tests

U03 L04

Absolute and Conditional Convergence

U03 L05

Alternating Series and Error Bounds

U03 L06

Power Series and Radius of Convergence

U03 L07

Operations on Power Series

U04 L01

Metrics and Open Balls

U04 L02

Open Sets, Closed Sets and Neighborhoods

U04 L03

Interior, Closure, Boundary and Accumulation Points

U04 L04

Convergence and Continuity in Metric Spaces

U04 L05

Completeness and Completion

U04 L06

Compactness and Sequential Compactness

U04 L07

Connectedness, Heine–Borel and Continuous Images

U05 L01

Epsilon–Delta Continuity

U05 L02

Sequential Characterization of Continuity

U05 L03

Algebra and Composition of Continuous Functions

U05 L04

Extreme and Intermediate Value Theorems

U05 L05

Uniform Continuity and Compact Domains

U05 L06

Function Spaces and the Supremum Norm

U05 L07

Equicontinuity and Arzelà–Ascoli Preview

U06 L01

Derivative as a Limit and Linear Approximation

U06 L02

Algebra of Derivatives and Chain Rule

U06 L03

Rolle’s Theorem and Mean Value Theorem

U06 L04

Consequences of the Mean Value Theorem

U06 L05

Taylor’s Theorem with Remainder

U06 L06

Convexity and Supporting Lines

U06 L07

Inverse and Implicit Function Preview

U07 L01

Partitions, Upper Sums and Lower Sums

U07 L02

Riemann Integrability Criteria

U07 L03

Continuous and Monotone Functions Are Integrable

U07 L04

Properties of the Riemann Integral

U07 L05

Fundamental Theorem of Calculus, Part I

U07 L06

Fundamental Theorem of Calculus, Part II and Substitution

U07 L07

Improper Integrals and Convergence

U08 L01

Pointwise versus Uniform Convergence

U08 L02

Uniform Cauchy Criterion

U08 L03

Weierstrass M-Test

U08 L04

Continuity under Uniform Limits

U08 L05

Interchanging Limits and Integrals

U08 L06

Interchanging Limits and Derivatives

U08 L07

Power-Series Uniformity and Analytic Preview

U09 L01

Sigma-Algebras and Measures

U09 L02

Outer Measure and Measurable Sets

U09 L03

Measurable Functions and Almost-Everywhere Statements

U09 L04

Lebesgue Integral of Simple and Nonnegative Functions

U09 L05

Integrable Functions and Comparison with Riemann Integration

U09 L06

Fatou, Monotone Convergence and Dominated Convergence

U09 L07

Lp Spaces, Hölder and Minkowski Inequalities

U10 L01

Complex Arithmetic, Geometry and Polar Form

U10 L02

Complex Exponential, Logarithm and Branches

U10 L03

Complex Limits and Continuity

U10 L04

Complex Differentiability and Holomorphic Functions

U10 L05

Cauchy–Riemann Equations

U10 L06

Harmonic Functions and Harmonic Conjugates

U10 L07

Conformal Maps and Möbius Transformations

U11 L01

Contours and Complex Line Integrals

U11 L02

Primitives and Path Independence

U11 L03

Cauchy–Goursat Theorem

U11 L04

Cauchy Integral Formula

U11 L05

Higher Derivatives and Cauchy Estimates

U11 L06

Liouville’s Theorem and Fundamental Theorem of Algebra

U11 L07

Maximum Modulus, Open Mapping and Identity Theorems

U12 L01

Laurent Series and Annuli of Convergence

U12 L02

Classification of Isolated Singularities

U12 L03

Residues and Coefficient Formulas

U12 L04

Residue Theorem

U12 L05

Real Integrals by Residues

U12 L06

Argument Principle and Rouché’s Theorem

U12 L07

Meromorphic Functions and Partial Fractions

U13 L01

Mean-Value Property and Poisson Kernel

U13 L02

Schwarz Lemma and Automorphisms of the Disk

U13 L03

Riemann Mapping Theorem: Statement and Consequences

U13 L04

Analytic Continuation and Monodromy Preview

U13 L05

Normal Families, Montel and Compactness Methods

U13 L06

Special Functions, Gamma and Zeta Previews

U13 L07

Research Problem Design and Literature Architecture

U13 L08

Expository Paper, Proof Defense and Graduate Bridge