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. What is sup {2 minus 1 over n, for n = 1, 2, ...}?

    Answer: ______________

  2. S equals {1 minus 1 over 2 to the n, for n = 1, 2, ...}. What is its supremum?

    • 0.5
    • 1, but it is not a maximum
    • 1, attained as a maximum by some term
    • 2

  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 positive integer n with 1 over n below 0.02.

    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. Working only inside the rationals, consider the set {q rational : q squared below 5}. What is its least upper bound?

    • root 5, which is not rational, is the real ceiling
    • No rational least upper bound exists
    • 5
    • 2

  8. If a set has a maximum, then its maximum equals its supremum.

    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. T equals (0, 2). What is its supremum?

    • 2, but it is not a maximum
    • 2, attained as a maximum at the endpoint
    • 0
    • There is no supremum
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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. 2 · The terms 1, 1.5, 1.666, and so on climb toward 2 without ever reaching it, so 2 is the least upper bound.
  2. 1, but it is not a maximum · The terms 0.5, 0.75, 0.875, and so on climb toward 1 without ever reaching it, so 1 is the least upper bound.
  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. 51 · 1 over n below 0.02 means n above 50, and n equal to 50 gives exactly 0.02, so the smallest positive integer n is 51.
  6. 5, which is also its maximum · The bound belongs to the set, so supremum and maximum coincide.
  7. No rational least upper bound exists · Over the rationals the only candidate ceiling is root 5, which is not rational, so no rational least upper bound exists.
  8. True · The maximum is an upper bound that belongs to the set, and any smaller bound would miss that member, so it is the least upper bound.
  9. True · The maximum does not exist there, so only the supremum can do the work.
  10. 2, but it is not a maximum · Every member sits below 2, any number below 2 is beaten by a member, and 2 lies outside the interval, so the supremum 2 is not a maximum.
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