The Completeness Axiom: Suprema and Infima · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Ceilings without a top step

Mathematics · Calculus & Analysis · ages 20-21
Name ______________________   Date ____________

  1. S = {0.9, 0.99, 0.999, ...}. What is sup S?

    Answer: ______________

  2. S = {1 - 1/n : n = 1, 2, ...}. What is its supremum?

    • 1, attained as a maximum
    • 0
    • 0.5
    • 1, but it is not a maximum

  3. S equals {1 minus 1 over n}. What is its supremum?

    • 0
    • 1, but it is not a maximum
    • 1, attained as a maximum

  4. S equals {0.9, 0.99, 0.999, ...}. What is sup S?

    Answer: ______________

  5. Find the smallest whole n with 1/n < 0.001.

    Answer: ______________

  6. S equals [2, 5]. What is its supremum?

    • 5, which is also its maximum
    • 2, attained at the left end
    • 5, which is not attained

  7. Why do the rationals fail the least upper bound property?

    • the rationals are countable
    • the rationals have no maximum
    • the set {q rational : q squared < 2} has no rational least upper bound
    • bounds only work for integers

  8. Using the supremum instead of the maximum is essential when the bound is not attained. True or false?

    Circle one:   True   False

  9. Using the supremum instead of the maximum is essential when the bound is not attained.

    Circle one:   True   False

  10. sup{x real : x squared < 9} equals what?

    • 3
    • 9
    • -3
    • 81
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Ceilings without a top step W1-mt_xjI-pIfh95-s1

  1. 1 · 1. The values close in on 1 from below without reaching it.
  2. 1, but it is not a maximum · 1, but it is not a maximum. Values climb toward 1 without ever reaching it.
  3. 1, but it is not a maximum · The terms climb toward 1 without ever reaching it.
  4. 1 · The terms climb toward 1 without reaching it.
  5. 1001 · 1001. Since 1/1000 equals 0.001 exactly, n must pass 1000.
  6. 5, which is also its maximum · The bound belongs to the set, so supremum and maximum coincide.
  7. the set {q rational : q squared < 2} has no rational least upper bound · The set {q rational : q squared < 2} has no rational least upper bound. Its real supremum root 2 is missing from Q.
  8. True · True. Maximums do not exist there, so arguments must cite the supremum.
  9. True · The maximum does not exist there, so only the supremum can do the work.
  10. 3 · 3. The set is (-3, 3), whose least upper bound is 3.
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