به انگلیسی

Ceilings without a top step

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

What is sup {2 minus 1 over n, for n = 1, 2, ...}?
Answer: ______________
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
S equals {1 minus 1 over n}. What is its supremum?
- 0
- 1, but it is not a maximum
- 1, attained as a maximum
S equals {0.9, 0.99, 0.999, ...}. What is sup S?
Answer: ______________
Find the smallest positive integer n with 1 over n below 0.02.
Answer: ______________
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
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
If a set has a maximum, then its maximum equals its supremum.
Circle one: True False
Using the supremum instead of the maximum is essential when the bound is not attained.
Circle one: True False
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

Answer key

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

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.
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.
1, but it is not a maximum · The terms climb toward 1 without ever reaching it.
1 · The terms climb toward 1 without reaching it.
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.
5, which is also its maximum · The bound belongs to the set, so supremum and maximum coincide.
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.
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.
True · The maximum does not exist there, so only the supremum can do the work.
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.