Math Ladder · College

Real Analysis

Why calculus works: the real numbers, limits with epsilons, continuity, derivatives, integrals, series of functions and metric spaces, proved carefully and tested with numbers.

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48 topics in 8 units. Builds on Calculus 2: Integrals and Series and Discrete Math. Not sure this is the right course? Find your rung.

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Unit 1 The Real Numbers

6 topics

Absolute value and the triangle inequality, suprema and infima, completeness, the density of the rationals, and the sizes of infinite sets.

  1. Absolute Value and the Triangle Inequality
  2. Supremum and Infimum
  3. The ε-Test for a Supremum
  4. Completeness and the Archimedean Property
  5. Density of the Rationals
  6. Countable and Uncountable Sets

Unit 2 Sequences and Their Limits

10 topics

The ε–N definition and proofs that use it, bounded and monotone sequences, subsequences, lim sup and lim inf, and Cauchy sequences.

  1. The Definition of a Limit (ε–N)
  2. Writing ε–N Proofs
  3. Bounded Sequences and Divergence
  4. Proving the Limit Laws
  5. Order Limits and the Squeeze Theorem
  6. The Monotone Convergence Theorem
  7. Subsequences and Subsequential Limits
  8. The Bolzano–Weierstrass Theorem
  9. Limit Superior and Limit Inferior
  10. Cauchy Sequences

Unit 3 Series

6 topics

Convergence as a statement about partial sums: the Cauchy criterion, comparison, condensation, the root test with lim sup, alternating series and rearrangements.

  1. Series and the Cauchy Criterion
  2. Comparison and Absolute Convergence
  3. The Cauchy Condensation Test
  4. Root and Ratio Tests with lim sup
  5. Alternating Series and Error Bounds
  6. Rearranging Series

Unit 4 Limits of Functions and Continuity

6 topics

Limits and continuity with ε and δ, the sequential criterion, the intermediate and extreme value theorems, and uniform continuity.

  1. Limits of Functions (ε–δ)
  2. Continuity with ε and δ
  3. The Sequential Criterion and Discontinuity
  4. The Intermediate Value Theorem
  5. The Extreme Value Theorem
  6. Uniform Continuity

Unit 5 Derivatives

5 topics

Derivatives from the definition, differentiability, the mean value theorem and the bounds it gives, and Taylor's theorem with remainder.

  1. Derivatives from the Definition
  2. Differentiability and Continuity
  3. Rolle's Theorem and the Mean Value Theorem
  4. Inequalities from the Mean Value Theorem
  5. Taylor's Theorem with Remainder

Unit 6 The Riemann Integral

6 topics

Upper and lower sums, the integrability criterion, which functions are integrable, bounds for integrals, and a proof of the fundamental theorem.

  1. Upper and Lower Sums
  2. The Integrability Criterion
  3. Which Functions Are Integrable
  4. Bounds and the Mean Value Theorem for Integrals
  5. Proving the Fundamental Theorem of Calculus
  6. Limits of Sums as Integrals

Unit 7 Sequences and Series of Functions

5 topics

Pointwise and uniform convergence, the sup norm, what uniform limits keep, the Weierstrass M-test and power series.

  1. Pointwise Convergence
  2. Uniform Convergence and the Sup Norm
  3. What Uniform Convergence Keeps
  4. The Weierstrass M-Test
  5. Power Series: Radius and Uniform Convergence

Unit 8 Metric Spaces

4 topics

Distance in general: metrics, open and closed sets, compact sets and the contraction mapping theorem.

  1. Metric Spaces
  2. Open and Closed Sets
  3. Compact Sets
  4. The Contraction Mapping Theorem
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