Tax, ox], is either empty, a single point, or a bounded closed interval. ΧΕΙ Exercise 2.1.7 a. Show that lim n = 0 (that is, the limit exists and is zero) if and only if n→∞0 lim n = 0. b. Find an example such that {n} converges and {n} diverges. Remarks

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Please solve Exercise 2.1.7... parts a and b 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).
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).
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