Does Scalar Multiplication Preserve the Supremum in Simple Sets?

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SUMMARY

The discussion confirms that for a set S defined as sup {Sn: n>N}, the equality kS = sup{kSn: n>N} holds true when k is any nonnegative scalar. Participants agree that the proof involves demonstrating that S serves as an upper bound and that no lower upper bound exists. This conclusion is established through logical reasoning and basic properties of supremum in mathematical analysis.

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  • Understanding of supremum and infimum concepts in real analysis
  • Familiarity with scalar multiplication in mathematical contexts
  • Basic knowledge of upper bounds and their properties
  • Experience with proof techniques in mathematics
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  • Study the properties of supremum and infimum in real analysis
  • Explore the implications of scalar multiplication on sets and their bounds
  • Learn about upper and lower bounds in mathematical proofs
  • Review examples of supremum preservation under various operations
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Mathematicians, students of real analysis, and anyone interested in the properties of supremum and scalar multiplication in mathematical sets.

Seacow1988
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If S=sup {Sn: n>N}, is it true that kS= sup{kSn: n>N} where k is any scalar?

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Woops. I meant k is any nonnegative scalar.

Thanks!
 
Yes, I think that this is indeed true for nonnegative scalars k. The proof is not so hard: just show that S is an upper bound and that no lower upper bound exists.
 

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