Testing is both technically and economically an important part of high quality software production. It has been estimated that testing accounts for half of the expenses in software production. Much of the testing is done manually or using other labor-intensive methods. It is thus vital for the software industry to develop efficient, cost effective, and automatic means and tools for software testing. Researchers have proposed several methods over years to generate automatically solution which have different drawbacks. This study examines automatic software testing optimization by using genetic algorithm approaches. This study will cover two approaches: a) obtain the sequence of regression tests that cover the greatest amount of code and b) once it is achieved another genetic algorithm will eliminate tests cases that cover the same section of code on the basis of still get the maximum code coverage. The overall aim of this research is to reduce the number of test cases that need to be run with the greatest amount of code covered.

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.