Exercise 1.4.6 a. Show that every closed interval [a, b] is the intersection of countably many open intervals. b. Show that every open interval (a, b) is a countable union of closed intervals. c. Show that an intersection of a possibly infinite family of bounded closed intervals, n[ax, bx], is either empty, a single point, or a bounded closed interval. ΧΕΙ
Exercise 1.4.6 a. Show that every closed interval [a, b] is the intersection of countably many open intervals. b. Show that every open interval (a, b) is a countable union of closed intervals. c. Show that an intersection of a possibly infinite family of bounded closed intervals, n[ax, bx], is either empty, a single point, or a bounded closed interval. ΧΕΙ
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please do the first exercise... parts a through c, with detailed explanations
![Exercise 1.4.6
a. Show that every closed interval [a, b] is the intersection of countably many open
intervals.
b. Show that every open interval (a, b) is a countable union of closed intervals.
c. Show that an intersection of a possibly infinite family of bounded closed intervals,
[ax, bx], is either empty, a single point, or a bounded closed interval.
λει
Exercise 2.1.7
a. Show that lim n = 0 (that is, the limit exists and is zero) if and only if
n→∞o
lim || = 0.
n∞0
b. Find an example such that {n} converges and {n}₁1 diverges.
Remarks
1
In parts (a) and (b) of Exercise 1.4.6, the intervals can be chosen to be nested.
In part (c) of Exercise 1.4.6, the conclusion is an "or" statement (in fancy language, a
disjunction). Parsing the logical structure reduces the problem to proving that if the
intersection contains at least two points, then the intersection must be a bounded closed
interval.
The point of part (b) of Exercise 2.1.7 is that the value 0 is special in part (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47d370c3-9e4b-442d-9a89-d591c5ced338%2F3f180005-df49-49f5-a125-6787ecafc4ba%2Fldaoz3b_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 1.4.6
a. Show that every closed interval [a, b] is the intersection of countably many open
intervals.
b. Show that every open interval (a, b) is a countable union of closed intervals.
c. Show that an intersection of a possibly infinite family of bounded closed intervals,
[ax, bx], is either empty, a single point, or a bounded closed interval.
λει
Exercise 2.1.7
a. Show that lim n = 0 (that is, the limit exists and is zero) if and only if
n→∞o
lim || = 0.
n∞0
b. Find an example such that {n} converges and {n}₁1 diverges.
Remarks
1
In parts (a) and (b) of Exercise 1.4.6, the intervals can be chosen to be nested.
In part (c) of Exercise 1.4.6, the conclusion is an "or" statement (in fancy language, a
disjunction). Parsing the logical structure reduces the problem to proving that if the
intersection contains at least two points, then the intersection must be a bounded closed
interval.
The point of part (b) of Exercise 2.1.7 is that the value 0 is special in part (a).
Expert Solution
Step 1
Exercise 1.4.6
(a) To Show: Every closed interval is the intersection of countable many open intervals.
(b) To Show: Every open interval is a countable union of closed intervals.
(c) To Show: An intersection of a possibly infinite family of bounded closed intervals, , is either empty, a single point, or a bounded closed interval.
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