Discussion Overview
The discussion revolves around proving the statement that the supremum of the union of two sets, sup(S ∪ T), equals the maximum of their individual suprema, max{supS, supT}. The focus is on the theoretical aspects of this proof without assuming any subset relationships between the sets S and T.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant requests assistance in completing or starting the proof, expressing frustration with the task.
- Another participant suggests beginning with the definition of the supremum of the union of two sets.
- A participant asserts that the problem is trivial, arguing that the largest number in the union must be the largest from either set, but acknowledges the complexity introduced by the concept of suprema.
- One participant expresses understanding of the answer but struggles with demonstrating that the maximum exists in the union and grapples with the idea that supT may not be contained within sup(S ∪ T).
- Another participant clarifies the definition of supremum and explains how it relates to the elements of the sets, suggesting that if L is the larger of the two suprema, it must also be the supremum of the union.
- There is a request for the original poster to attempt the reverse direction of the proof.
Areas of Agreement / Disagreement
Participants exhibit a mix of agreement on the definitions and intuitive understanding of the problem, but there is no consensus on how to effectively demonstrate the proof. Disagreement exists regarding the perceived difficulty of the proof and the clarity of the definitions involved.
Contextual Notes
Participants highlight the importance of definitions and the existence of elements in the context of suprema, but there are unresolved aspects regarding the completeness of the proof and the assumptions made about the sets.