Proving Unboundedness of B if A is Unbounded

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SUMMARY

This discussion addresses the mathematical proof that if set A is an unbounded subset of set B, then B must also be unbounded. The argument begins by assuming the contrary—that B is bounded, which leads to a contradiction when considering the definition of boundedness. The conclusion is that the properties of subsets directly influence the boundedness of larger sets, making the proof straightforward for those familiar with the definitions involved.

PREREQUISITES
  • Understanding of real number sets and their properties
  • Familiarity with the concepts of boundedness in mathematics
  • Knowledge of set theory, particularly subsets
  • Basic mathematical proof techniques
NEXT STEPS
  • Study the definitions and properties of bounded and unbounded sets in real analysis
  • Explore advanced topics in set theory, focusing on subsets and their implications
  • Review mathematical proof strategies, particularly proof by contradiction
  • Investigate related concepts such as limits and convergence in real analysis
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Mathematicians, students of real analysis, and anyone interested in the foundational concepts of set theory and boundedness in mathematics.

Seacow1988
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If A, B are nonempty subsets of R and A is a subset of B, how can you prove that: if A is unbounded, B is unbounded?
 
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Suppose to the contrary that B is bounded. So there exists a real number M such that |b| < M for all b in B. Do you see where the problem lies?
 
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If you understand the definition of 'boundedness' and 'subset' then this is trivial.
 

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