Real Analysis

The proofs behind the calculus you already know

About this course

A rigorous development of the real line: completeness, sequences, continuity, differentiation, the Riemann integral and uniform convergence.

What you will be able to do

  • Use the completeness of the reals and the epsilon-N definition of a limit
  • Prove continuity results, including uniform continuity
  • Prove the mean value theorem and Taylor's theorem with remainder
  • Construct the Riemann integral and state where it fails
  • Distinguish pointwise from uniform convergence and know why it matters

What this course covers

  • The derivative as a limit
  • Linear-approximation reformulation of differentiability
  • Differentiable implies continuous (and the hierarchy of smoothness)
  • Weierstrass's nowhere-differentiable function
  • Positive derivative at a point does not imply locally increasing
  • Darboux's theorem: derivatives have the intermediate value property
  • Working rules, proof strategies and standard errors
  • Pointwise vs. uniform convergence: quantifier order
  • What pointwise convergence fails to preserve
  • Uniform convergence: definition, sup-norm, and Cauchy criterion
  • What uniform convergence buys: continuity, integral, completeness
  • Uniform convergence does not control derivatives
  • Weierstrass's function as a uniform limit of smooth functions
  • Dini's theorem: uniform convergence for free
  • Partitions, upper and lower sums, and the squeeze definition
  • Worked example: integral of x² from 0 to 1
  • Non-integrability of Dirichlet's function
  • Riemann's criterion and integrability of continuous functions
  • Integrability of monotone functions by telescoping
  • Thomae's function and measure-zero sets
  • Lebesgue's criterion: the exact characterisation of Riemann integrability
  • The defect of the rational numbers
  • Supremum and infimum
  • The completeness axiom and existence of √2
  • Two-step supremum proof technique and proof habits
  • The Archimedean property and density of the rationals
  • Why the reals matter for computation
  • Cantor's diagonal argument
  • Limitations of the intuitive continuity picture
  • Epsilon-delta definition of continuity at a point
  • Sequential characterisation of continuity
  • Pathological functions that cannot be drawn
  • Which sets can be sets of continuity (G-delta sets)
  • Summary: toolkit, standard errors, and course spine
  • Worked epsilon-delta proof for 1/x at a = 2
  • Why a bridge from points to intervals is needed
  • Rolle's Theorem and its proof
  • The Mean Value Theorem and Cauchy's generalisation
  • The four-line technique for applying the MVT
  • Harvest: four corollaries of the MVT
  • L'Hôpital's rule: honest statement and failure modes
  • Worked example: proving ln(1+x) < x via MVT
  • Epsilon-N definition of convergence
  • Negation of convergence
  • Historical context: Weierstrass and rigorous analysis
  • Uniqueness of limits and the halving maneuver
  • Algebra of limits and the squeeze theorem
  • Monotone convergence theorem
  • Bounded alone is insufficient; divergent sequence examples
  • Existence theorems vs. descriptive theorems
  • Intermediate Value Theorem: statement and proof
  • Applications of the IVT: fixed points, roots, antipodal temperatures
  • IVT fails over the rationals: completeness is essential
  • Extreme Value Theorem: statement and proof
  • EVT fails over the rationals; compactness foreshadowed
  • The sorting rule: completeness manufactures points
  • Two Theorems, Not One
  • Part 1: Building an Antiderivative from an Integral
  • Historical Attribution of the Fundamental Theorem
  • Part 2: Evaluating an Integral with an Antiderivative
  • Why the Two Theorems Are Genuinely Separate
  • Almost Everywhere: The General Modern Statement
  • Working Procedures, Standard Errors, and the Notation Warning
  • Weierstrass M-test
  • Power series and radius of convergence
  • Term-by-term differentiation and integration theorems
  • What first-year calculus was justified in doing
  • Failure of term-by-term differentiation for general series
  • Weierstrass approximation theorem and Bernstein's proof
  • Course synthesis: hypothesis, counterexample, and method
  • Limits of existing tools and their gaps
  • Subsequences: definition and index lemma
  • Bolzano–Weierstrass theorem
  • Subsequential limits, limsup and liminf
  • Cauchy sequences: definition and equivalence to convergence
  • Harmonic sums: consecutive differences vs. Cauchy
  • Completeness: equivalent properties and the Cauchy criterion's place
  • Quantifier swap: uniform vs pointwise continuity
  • Negation and working tool for disproving uniform continuity
  • Failures of uniform continuity: three worked examples
  • Heine–Cantor theorem and its two proofs
  • Chain of implications: Lipschitz → uniform continuity → continuity
  • Applications: Riemann integral and Cauchy's false theorem
  • Worked example: √x on [0,1] and on [1,∞)
  • Built-in limits of Riemann's definition
  • Failure of Riemann integrability under pointwise limits
  • Failure of integral–limit interchange and uniform convergence
  • Incompleteness of Riemann integrable functions in L1
  • Volterra's function: bounded derivative not Riemann integrable
  • The Smith–Volterra–Cantor set
  • Lebesgue integration: partitioning the range
  • Taylor Polynomial: Definition and Uniqueness
  • Best-Approximation Sense and Three Separate Statements
  • Lagrange Remainder: Statement and Proof Sketch
  • Remainder Vanishing as the Criterion for Series Equality
  • Cauchy's Smooth Non-Analytic Function
  • Radius of Convergence and Complex Singularities
  • Worked Example and Practical Procedures

Syllabus

  1. The Real Line and Sequences

    1. L1 · Why the Rationals Are Not Enough
    2. L2 · Sequences and the Epsilon-N Definition
    3. L3 · Bolzano-Weierstrass, and Cauchy

  2. Continuity

    1. L4 · The Epsilon-Delta Definition
    2. L5 · What Continuity Buys
    3. L6 · Uniform Continuity

  3. Differentiation

    1. L7 · The Derivative, Made Rigorous
    2. L8 · The Mean Value Theorem
    3. L9 · Taylor's Theorem, with the Remainder

  4. Integration

    1. L10 · The Riemann Integral
    2. L11 · The Fundamental Theorem, with Hypotheses
    3. L12 · Where Riemann Runs Out

  5. Convergence of Functions

    1. L13 · Pointwise Versus Uniform
    2. L14 · Series of Functions, and What It All Justified