Would it be true that if a set is bounded

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SUMMARY

If a set is bounded and consists of real numbers, it possesses a supremum, which is a defining property of the real numbers. However, this does not hold true for all sets; for instance, the set of rational numbers where x^2 < 2 is bounded but lacks a rational supremum, as its supremum is the irrational number √2. The discussion emphasizes the importance of defining the type of set and the metric involved when considering boundedness and supremum.

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  • Understanding of real numbers and their properties
  • Familiarity with rational numbers and their limitations
  • Basic knowledge of set theory and bounded sets
  • Concept of supremum and infimum in mathematical analysis
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  • Study the properties of real numbers and their completeness
  • Examine the concept of supremum in different number systems, including rational and irrational numbers
  • Explore set theory fundamentals, focusing on bounded and unbounded sets
  • Learn about metrics and their role in defining boundedness in various mathematical contexts
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Mathematicians, students of advanced mathematics, and anyone interested in the properties of sets and number systems will benefit from this discussion.

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In general, would it be true that if a set is bounded, there must also be a supremum for the set? Too obvious, perhaps?
 
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DPMachine said:
In general, would it be true that if a set is bounded, there must also be a supremum for the set? Too obvious, perhaps?
Define your terms! A "set" does not even have to consist of numbers. In order to talk about a set being "bounded" there must be some kind of metric defined on it but even then there may not be an order- the set of all complex numbers with norm less than 1 is bounded but is not an ordered set and so "supremum" makes no sense.

I assume you are talking about a set of real numbers. Yes, any set of real numbers, having an upper bound, has a real number as supremum. That's one of the "defining" properties of the real numbers.

It is NOT true, for example, if you are talking about rational numbers. For example, the set of all rational numbers, x, such that [math]x^2< 2[/math] has such numbers as 1.5, 2, etc. as upper bounds but does not have a rational number as supremum. As a set of real numbers, its supremum is \sqrt{2}.
 

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