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.
EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!
E-mail: easydue@outlook.com 微信:easydue
EasyDue™是一个服务全球中国留学生的专业代写公司
专注提供稳定可靠的北美、澳洲、英国代写服务
专注提供CS、统计、金融、经济、数学等覆盖100+专业的作业代写服务