Generalization of Exercise 2.3.8. Suppose {xn}~_=1 and {yn}_₁ are two bounded sequences of real numbers. a. Prove that lim sup(x₂ + Yn) ≤(lim sup än) +(lim sup yn Yn), n→∞ n→∞ n→∞ and give an example to show that strict inequality can occur. B. Prove that if the numbers an and yn are positive for every n, then lim sup(nyn) ≤(lim sup xn) (lim sup yn Yn), n→∞ n→∞ n→∞ and give an example to show that the inequality can fail when the positivity hypothesis is omitted. Exercise 2.4.4. Let {xn}a_1 and {yn}=1 be sequences of real numbers such that limn→∞ Yn = 0. Suppose for every natural number k, if m ≥ k then |ïm − xk| ≤ Yk. Prove that {n}_1 is a Cauchy sequence. marks -
Generalization of Exercise 2.3.8. Suppose {xn}~_=1 and {yn}_₁ are two bounded sequences of real numbers. a. Prove that lim sup(x₂ + Yn) ≤(lim sup än) +(lim sup yn Yn), n→∞ n→∞ n→∞ and give an example to show that strict inequality can occur. B. Prove that if the numbers an and yn are positive for every n, then lim sup(nyn) ≤(lim sup xn) (lim sup yn Yn), n→∞ n→∞ n→∞ and give an example to show that the inequality can fail when the positivity hypothesis is omitted. Exercise 2.4.4. Let {xn}a_1 and {yn}=1 be sequences of real numbers such that limn→∞ Yn = 0. Suppose for every natural number k, if m ≥ k then |ïm − xk| ≤ Yk. Prove that {n}_1 is a Cauchy sequence. marks -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve exercise 2.3.8 with detailed explanations for each step. Thank you
Transcribed Image Text:• Generalization of Exercise 2.3.8. Suppose {n}a_1 and {yn}_1 are two bounded sequences of real numbers.
a. Prove that
lim sup(n + yn) ≤ (lim sup n) + (lim sup yn),
n→∞
n→∞
n→∞
and give an example to show that strict inequality can occur.
B. Prove that if the numbers in and yn are positive for every n, then
lim sup(nyn) ≤(lim sup xn) (lim sup yn),
n→∞
n→→∞
n→∞
and give an example to show that the inequality can fail when the positivity hypothesis is omitted.
=
• Exercise 2.4.4. Let {xn}_1 and {Yn}~1 be sequences of real numbers such that limn→∞ Yn 0. Suppose that
for every natural number k, if m ≥ k then xm xk| ≤ Yk. Prove that {n}_1 is a Cauchy sequence.
Remarks
Proposition 2.2.5 says that the limit operation respects the field operations. The operation of limit superior, however,
does not preserve the field operations. Exercise 2.3.8 shows that something can be salvaged. As usual when dealing
with the notion of lim sup, establishing the inequality up to an arbitrarily small positive error is sufficient.
The hypothesis of Exercise 2.4.4 entails that yk ≥ 0 for every k.
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