Let (xn) be a bounded sequence. Recall that lim inf xn = sup inf xm = sup inf{rn, Xn+1, ...}, lim sup , = inf sup am = inf sup{xn, Cn+1, - .}. +1,•. m>n n m>n n00 n00 n Show that L = lim inf xn is the smallest limit point of (xn), i.e., any subsequential limit of (xn) is not smaller n00 than L, and there is a subsequence that has L as its limit.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

How do you solve the first bullet point?

Let (xn) be a bounded sequence. Recall that
lim inf xn = sup inf xm =
sup inf{xn, &n+1,..}, lim sup an = inf sup am = infsup{rn, &n+1,• .}.
m>n
n m>n
n00
Show that L = lim inf xn is the smallest limit point of (xn), i.e., any subsequential limit of (xn) is not smaller
n-00
than L, and there is a subsequence that has L as its limit.
Remark: Similarly, lim sup xn is the largest limit point of (xn). This also shows that a sequence (xn) converges if
n00
and only if its lim sup and lim inf are equal.
• Let (xn) and (Yn) be bounded sequences and xn < Yn. Show that
lim inf xn <lim inf yn .
n00
n00
• Let (xn) and (Yn) be bounded sequences. Show that
lim inf xn + lim inf yn < lim inf(xn + Yn) < lim inf xn + lim sup Yn,
n00
n00
n00
n00
n00
Hint: for parts 2) and 3), you can take some proper subsequences and use part 1).}
Transcribed Image Text:Let (xn) be a bounded sequence. Recall that lim inf xn = sup inf xm = sup inf{xn, &n+1,..}, lim sup an = inf sup am = infsup{rn, &n+1,• .}. m>n n m>n n00 Show that L = lim inf xn is the smallest limit point of (xn), i.e., any subsequential limit of (xn) is not smaller n-00 than L, and there is a subsequence that has L as its limit. Remark: Similarly, lim sup xn is the largest limit point of (xn). This also shows that a sequence (xn) converges if n00 and only if its lim sup and lim inf are equal. • Let (xn) and (Yn) be bounded sequences and xn < Yn. Show that lim inf xn <lim inf yn . n00 n00 • Let (xn) and (Yn) be bounded sequences. Show that lim inf xn + lim inf yn < lim inf(xn + Yn) < lim inf xn + lim sup Yn, n00 n00 n00 n00 n00 Hint: for parts 2) and 3), you can take some proper subsequences and use part 1).}
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,