# Course Description

This course was taught by professor Raymond Deneckere. The first half of the course was taught during the summer and the second half during the fall semester. The intended audience is first-year Ph.D. students in the economics department.

In this proof-based course, students develop the tools to understand and formulate rigorous mathematical proofs. They learn tools in real analysis to prove classic theorems like Weierstrass, the implicit function theorem and optimization theorems (Lagrange and Kuhn Tucker).

# Material

Topics | Handout | Solutions | |
---|---|---|---|

Discussion 1 | Sets and Negations | Handout 1 | Solutions 1 |

Discussion 2 | Induction, functions and general relations | Handout 2 | Solutions 2 |

Discussion 3 | Supremums, sequences, limits and vector spaces | Handout 3 | Solutions 3 |

Discussion 4 | Metric Spaces and Topology | Handout 4 | Solutions 4 |

Discussion 5 | Limit points and compact sets | Handout 5 | Solutions 5 |

Discussion 6 | Continuity of functions | Handout 6 | Solutions 6 |

Discussion 7 | Differentiability, IVT and MVT | Handout 7 | Solutions 7 |

Discussion 8 | Multivariate calculus and optimization | Handout 8 | Solutions 8 |

Discussion 9 | Complete spaces | Handout 9 | Solutions 9 |

Discussion 10 | Contractions and IFT | Handout 10 | Solutions 10 |

Discussion 11 | Hyperplanes and constrained optimization | Handout 11 | Solutions 11 |

Discussion 12 | Review | Handout 12 | Solutions 12 |

Discussion 13 | Constrained and unconstrained optimization | Handout 13 | Solution 13 |

Discussion 14 | Constrained optimization and convex sets | Handout 14 | Solution 14 |

Discussion 15 | Concave functions and convex optimization | Handout 15 | Solution 15 |

Discussion 16 | Quasi-concavity | Handout 16 | Solution 16 |

Discussion 17 | Correspondences and the theorem of the maximum | Handout 17 | Solution 17 |

#### Disclaimer

- Some of the exercises are inspired in published work. The selection of questions was meant to assist students in their learning process. I do not claim property rights of the ideas presented.
- I appreciate the collaboration of former students in identifying mistakes in the problems or solutions. I constantly keep them updated. If you find an error, please contact me.