Demonstration of Theorem on Compact Functions

This lecture covers a detailed demonstration of a theorem related to continuous functions on a compact set, focusing on the proof that a function attains its minimum and maximum. The instructor explains the concept of boundedness on a compact set, the distinction between sequences of function values and points, and the convergence of subsequences. The lecture also delves into the properties of supremum and infimum values, using the definitions to show convergence. Additionally, an example of a non-regular boundary curve is presented, illustrating a construction method that results in a continuous curve without self-intersections. The lecture concludes with insights on solving systems of equations, analyzing limits, and determining open and closed sets in relation to compactness.