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Mathematics LibreTexts

7.E: Intermediate and Extreme Values (Exercises)

 

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Q1

Mimic the definitions of an upper bound of a set and the least upper bound (supremum) of a set to give definitions for a lower bound of a set and the greatest lower bound (infimum) of a set.

Note: The infimum of a set \(S\) is denoted by \(\inf (S)\).

Q2

Find the least upper bound (supremum) and greatest lower bound (infimum) of the following sets of real numbers, if they exist. (If one does not exist then say so.)

  1. \(S = \{\frac{1}{n} |n = 1,2,3,...\} \)
  2. \(T = \{r|r \text{ is rational and }r^2 < 2\}\)
  3. \((-∞,0) ∪ (1,∞)\)
  4. \(R = \{\frac{(-1)^n}{n} |n = 1,2,3,...\}\)
  5. \((2,3π] ∩ \mathbb{Q}\)
  6. The empty set \(\varnothing\) 

Q3

Let \(S ⊆ R\) and let \(T = \{-x|x ∈ S\}\).

  1. Prove that \(b\) is an upper bound of \(S\) if and only if \(-b\) is a lower bound of \(T\).
  2.  Prove that \(b = \sup S\) if and only if \(-b = \inf T\).

Contributor

  • Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)