#### Degree Name

MS (Master of Science)

#### Program

Mathematical Sciences

#### Date of Award

5-2021

#### Committee Chair or Co-Chairs

Anant Godbole.

#### Committee Members

Robert B. Gardner, Rodney L. Keaton

#### Abstract

This thesis takes a closer look at three fundamental Classical Theorems in Real Analysis. First, for the Bolzano Weierstrass Theorem, we will be interested in constructing a convergent subsequence from a non-convergent bounded sequence. Such a subsequence is guaranteed to exist, but it is often not obvious what it is, e.g., if an = sin n. Next, the H¨older Inequality gives an upper bound, in terms of p ∈ [1,∞], for the the integral of the product of two functions. We will find the value of p that gives the best (smallest) upper-bound, focusing on the Beta and Gamma integrals. Finally, for the Weierstrass Polynomial Approximation, we will find the degree of the approximating polynomial for a variety of functions. We choose examples in which the approximating polynomial does far worse than the Taylor polynomial, but also work with continuous non-differentiable functions for which a Taylor expansion is impossible.

#### Document Type

Dissertation - unrestricted

#### Recommended Citation

Elallam, Abderrahim, "Constructions & Optimization in Classical Real Analysis Theorems" (2021). *Electronic Theses and Dissertations.* Paper 3901. https://dc.etsu.edu/etd/3901

#### Copyright

Copyright by the authors.