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johnqwertyful
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NATURE.M said:
No more than you need to justify n+1>n.
Explaining Why 1.1 and 0.95 Are Not Least Upper Bounds for Set A
- Thread starter NATURE.M
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SUMMARY
The discussion centers on proving that 1.1 and 0.95 are not the least upper bounds (LUB) for the set A, defined as all numbers of the form n/(n+1) where n is a positive integer. Participants clarify that 1.1 cannot be the LUB because there exists an upper bound, such as 1.01, that is less than 1.1, violating the LUB definition. Similarly, 0.95 is not the LUB since numbers like 1/2 (0.5) exist in set A that are greater than 0.95, confirming that 0.95 does not satisfy the upper bound condition.
PREREQUISITES- Understanding of the least upper bound axiom in real analysis.
- Familiarity with the concept of upper bounds in mathematical sets.
- Basic knowledge of limits and sequences, particularly the behavior of n/(n+1) as n approaches infinity.
- Ability to construct mathematical proofs and counterexamples.
- Study the properties of bounded sets in real analysis.
- Learn how to construct proofs using the least upper bound axiom.
- Explore examples of sequences and their limits, particularly focusing on rational functions.
- Practice writing concise mathematical proofs to improve clarity and effectiveness.
Students of mathematics, particularly those studying real analysis, educators teaching proof construction, and anyone interested in understanding upper bounds and limits in mathematical sets.
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