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Worksheet: Bounded Sequences

Q1:

Which of the following describes the least upper bound β„“ of a subset 𝑆 of numbers?

  • Athe largest number amongst all 𝑠 ∈ 𝑆
  • Bthe number β„“ so that if 𝑠 ∈ 𝑆 , then β„“ β‰₯ 𝑠
  • Cthe smallest number amongst all 𝑠 ∈ 𝑆
  • Dthe number β„“ so that if 𝑏 β‰₯ 𝑠 for all 𝑠 ∈ 𝑆 , then β„“ ≀ 𝑏
  • Ethe number β„“ so that if 𝑏 ≀ 𝑠 for all 𝑠 ∈ 𝑆 , then β„“ β‰₯ 𝑏

Q2:

Suppose 𝑆 is a subset of real numbers that has an upper bound 𝑏 .

Suppose that β„“  is a least upper bound of 𝑆 . Which of the following relations between β„“  and 𝑏 must be true?

  • A β„“ = 𝑏 
  • B β„“ < 𝑏 
  • C β„“ β‰  𝑏 
  • D β„“ ≀ 𝑏 
  • E β„“ β‰₯ 𝑏 

Suppose β„“  is a second least upper bound for 𝑆 . Which of the following relations between β„“  and β„“  must be true?

  • A β„“ > β„“  
  • B β„“ < β„“  
  • C β„“ β‰  β„“  
  • D β„“ = β„“  