这个作业是证明上限下限值以及无限逼近值相关的数学问题
Math 555, Spring 2021
Homework #7
EX.1: For each of the following functions, decide which are bounded above or bounded
below on the indicated interval, and which take on their maximum or minimum value. Be
sure to include explanations.
(a) f(x) = x
2 on (−1, 1).
(b) f(x) = x
3 on (−1, 1).
(c)
f(x) = (
0 when x 6∈ Q
1
q
when x ∈ Q, x =
p
q
in lowest terms
on (0, 1).
EX.2: Prove that a function f is bounded on [a, b] if and only if |f| is bounded on [a, b].
EX.3: Let f be continuous on [0,∞), and suppose limx→∞ f(x) = L exists. Prove that f
is bounded on [0, ∞). (Hint: Find N such that when x > N, f(x) is close to L. Then you
can apply the awesome theorems to the interval [0, N].
Ex.4: Give counterexamples for the following statements.
(a) If A is any bounded subset of R and α = sup A, then for any > 0, (α − , α) ∩ A 6= ∅.
(b) If A is any bounded subset of R and β = inf A, then for any > 0, (β, β + ) ∩ A 6= ∅.
EX.5: Prove the following facts about sup and inf. (All the proofs should follow directly
from the definition. The statements are fairly intuitive and you may find it helpful to draw
a picture.) Let A and B be bounded subsets of R.
(a) If c ∈ R, we define cA = {ca : a ∈ A}. Prove that inf cA is equal to c · inf A if c ≥ 0
and c · sup A if c < 0. State a similar result for sup cA (you don’t have to prove it).
(b) Suppose A ⊂ B. Prove that inf A ≥ inf B. State a similar result for sup A and sup B
(you don’t have to prove it).
(c) Suppose a ≤ b for any a ∈ A and b ∈ B. Prove that sup A ≤ inf B.
(d) Suppose sup A ≤ inf B, and for any > 0, there exists a ∈ A and b ∈ B such that
b − a ≤ . Prove that sup A = inf B.