The discussion revolves around proving that an even degree polynomial has either an absolute maximum or minimum. Participants explore the properties of polynomials, particularly their behavior at infinity, and the implications of continuity and boundedness in relation to extreme values.
The discussion is active, with participants providing insights and alternative perspectives on the proof. Some express satisfaction with the reasoning presented, while others seek simpler or more elegant proofs. There is an ongoing examination of the relationships between different mathematical theorems and concepts, such as the Extreme Value Theorem and Bolzano-Weierstrass.
Participants note the importance of assumptions regarding the leading coefficient and the behavior of the polynomial at both ends of the x-axis. There is a focus on the implications of these assumptions for the sets defined in the proof.
Dick said:I just meant that the guts of the proof of the EVT is BW added to the definition of continuity. That's all. Not that they are somehow formally equivalent. And yes, your proof was on the right track. I was thinking EVT was the continuous function bounded theorem. Just confused.