We will form a proof of the Arzela-Ascoli Theorem through use of the Heine-Borel theorem. We will also be considering some notions of compactness on metric spaces. The Arzela-Ascoli Theorem then allows us to show compactness, letting us state and prove Peano's existence theorem, pertaining to the existence of the solutions of a type of ODE. Then we will state the Kolmogorov-Riesz compactness theorem, allowing us to show compactness in $L^p$ spaces, building from the Arzela-Ascoli Theorem.
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Beginning template for the 2015 Mathematical Contest in Modeling, http://www.comap.com/undergraduate/contests/.
This template will place the correct header on each page and reserves the first page for the required model summary.
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