MHB Is Max A $\le$ Sup B for A $\subseteq$ B?

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The discussion centers on whether the maximum of set A, denoted as max A, is less than or equal to the supremum of set B, given that A is a subset of B. It questions the existence of a maximum for set A and asserts that if A is indeed a subset of B, then the supremum of A is always less than or equal to the supremum of B. The relationship between the maximum and supremum in this context is explored, emphasizing the importance of A's properties. Overall, the conclusion highlights that while max A may not always exist, the inequality sup A ≤ sup B holds true. The discussion reinforces key concepts in set theory and order relations.
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for A$\subset$ B max A $\le$ sup B ? is it true ?
 
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Is the maximum defined for $A$? Regardless, it is true that if $A\subseteq B$, then $\sup A\le \sup B$.
 

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