# Real analysis

Chapter and section references are from Corbae, Stinchcombe and Juraj (2009), “An Introduction to Mathematical Analysis for Economic Theory and Econometrics.'' I only recommend you follow Corbae et al. if you are already comfortable with highly-technical math. Otherwise, use this list of topics as guidance while using one of the recommended sources on the Textbooks page.

## Logic

Chapter 1 in Corbae, Stinchcombe and Juraj (2009)

Topic | Reference | Background | Proofs | Direct/General |
---|---|---|---|---|

Statements, Sets, Subsets, Implication | Section 1.1 | X | X | X |

Ands, Ors, Nots | Section 1.2.a | X | X | X |

Implies, Equivalence | Section 1.2.b | X | X | X |

Vacuous Statements | Section 1.2.c | X | X | X |

Indicators | Section 1.2.d | X | X | |

Logical Quantifiers | Section 1.4 | X | X | |

Taxonomy of Proofs | Section 1.5 | X | X | X |

## Set Theory

Chapter 2 in Stinchcombe and Juraj (2009)

Topic | Reference | Background | Proofs | Direct/General |
---|---|---|---|---|

Notation for sets | 2.2.2, top pg. 21 | X | X | |

Useful theorems on sets | 2.2.4, 2.2.6 | X? | ||

Cartesian Product | 2.3.1 | X | X | X |

Relation | 2.3.4, 2.3.5 | X | ||

Function | 2.3.8 | X | X | X |

Correspondence | 2.3.12 | X | X | X |

Image | 2.3.16, 2.6.1, 2.6.4 | X | X | |

Cardinality | 2.3.17 | X | X | X |

Equivalence Class | 2.4.1, 2.4.5 | X | X | X |

Partition | 2.4.9 | X | X? | |

Inverse, Inverse Image | 2.6.7, 2.6.10, 2.6.13 | X | X | X |

Level Sets of Functions | 2.6.12 | X | X | X |

One-to-One / Injection | 2.6.15 | X | X | |

Onto / Surjection / Bijection | 2.6.17 | X | X | |

Composite Functions | 2.6.20, 2.6.23, 2.6.26 | X | X |

## The Space of Real Numbers

Chapter 3 in Stinchcombe and Juraj (2009)

Topic | Reference | Background | Proofs | Direct/General |
---|---|---|---|---|

The `Why' | Section 3.1 and 3.10 | X | ||

Algebraic Properties of $\mathbb{Q}$ | 3.2.3 | X | X | |

Distance in $\mathbb{Q}$ | 3.3.1, 3.3.2 | X | X | |

Sequence | 3.3.3 | X | X | X |

Subsequence | 3.3.5 | X | X | |

Cauchy | 3.3.7, 3.4.8 | X | ||

Bounded | 3.3.12, 3.3.13, 3.6.1 | X | X | X |

Real Numbers | 3.3.19 | X | ||

Algebraic Properties of $\mathbb{R}$ | 3.3.23 | X | X | |

Distance in $\mathbb{R}$ | 3.4.1, 3.4.2, 3.4.3 | X | X | |

Convergence | 3.4.9, 3.4.10, 3.4.15 | X | X | X |

Completeness of $\mathbb{R}$ | 3.4.16 | X | ||

Supremum / Infimum | 3.6.2, 3.6.5 | X | X | X |

## The Finite-Dimensional Metric Space of Real Vectors

Chapter 4 in Stinchcombe and Juraj (2009)

Topic | Reference | Background | Proofs | Direct/General |
---|---|---|---|---|

Metric Space | 4.1.1 | X | ||

Convergence, Limit | 4.1.4 | X | X | X |

Complete | 4.1.6, 4.4.5 | X | X | |

Open ball | 4.1.9 | X | ||

Open | 4.1.10, 4.1.11 | X | X | |

Open neighborhood | 4.1.12 | X | ||

Open cover | 4.1.18 | X | ||

Compact | 4.1.19, 4.7.15 | X | X | |

Connected | 4.1.21 | X | X | |

Continuous | 4.1.22, 4.7.20, 4.85 | X | X | X |

Vector Space | 4.3.1 | X? | X | |

Normed Vector Space | 4.3.7 | X? | ||

Inner / Dot Product | 4.3.9 | X | X | X |

Cauchy-Schwartz Inequality | 4.3.10 | X | ||

\textit{p}-Norms | Section 4.3.c | X? | X | |

Characterizing Closed Sets | Section 4.5.a | X | X | |

Closure of a Set | 4.5.4 | X | ||

Boundary of a Set | 4.5.5 | X | X | |

Accumulation / Cluster / Limit Point | 4.5.7 | X | X | |

Closure and Completeness | 4.5.12, 4.5.13 | X | X | |

Bounded | 4.7.8 | X | X | X |

Applications of Compactness | Section 4.7.f | X | X | X |

Basic Existence Result | 4.8.11, 4.8.16 | X | X | |

Upper Hemicontinuity | 4.10.20 | X | X | |

Theorem of the Maximum | 4.10.22, 6.1.31 | X | X | |

Upper Semicontinuity | 4.10.29 | X | X? | |

Connected | 4.1.12, 4.12.3, 4.12.4 | X | X | X |

Interval | 4.12.1, 4.12.2 | X | X | X |

Intermediate Value Theorem | 4.12.5 | X | X | X |

## Finite-Dimensional Convex Analysis

Chapter 5 in Stinchcombe and Juraj (2009)

Topic | Reference | Background | Proofs | Direct/General |
---|---|---|---|---|

Convex Combination | 5.1.2 | X | X | X |

Convex Preferences and Technologies | Section 5.1.c | X | ||

Returns to Scale | 5.1.13 | X | ||

Convex Hull | 5.4.6 | X | ||

Upper Contour Set | 5.4.23 | X | X | X |

Affine Combination | 5.6.16 | X | ||

Interior | 5.5.1, 5.5.2 | X | ||

Concave Function | 5.6.1, 5.6.2 | X | X | X |

Affine Function | 5.6.6 | X | X | |

Quasi-Concave | 5.6.12 | X | X | X |

Single-Peaked | 5.6.13 | X | ||

Implicit Function Theorem | Sections 2.8.a and 5.9.b | X | X | |

Envelope Theorem | Section 5.9.c | X | X |

Sections 5.8 - 5.10 contain results on optimization. The facts and results that you need should already be familiar from math camp so I do not list them separately.