Real Analysis, practice ex 1 Exercise 0.1.Assume that (xn) is a bounded sequence.3(a) Prove that lim inf(x) = (lim inf xn)°.(b) Is the statement lim inf(x) = (lim inf xn)² true in general?

Answered: Exercise 0.1. Assume that (xn) is a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Real Analysis, practice ex 1

Exercise 0.1.
Assume that (xn) is a bounded sequence.
3
(a) Prove that lim inf(x) = (lim inf xn)°.
(b) Is the statement lim inf(x) = (lim inf xn)² true in general?

Expert Solution

Step 1

Given ${x_{n}}$ be a bounded sequence

a) we have to show that $l i m i n f \{{x^{3}}_{n}\} = {\{l i m i n f x_{n}\}}^{3}$

now let ${x_{n}} = {(- 1)}^{n}$ be a bounded sequence

${(x_{n})}^{3} = {{(- 1)}^{n}}^{3} = {(- 1)}^{3 n} {(x_{n})}^{3} = {- 1, 1, - 1, 1, - 1, 1 . . . . . . . .} t h e n l i m {(x_{n})}^{3} = {- 1, 1} \Rightarrow l i m i n f {(x_{n})}^{3} = - 1 a n d x_{n} = {- 1, 1, - 1, 1, - 1, 1 . . . . . .} t h e n l i m x_{n} = < - 1, 1 > \Rightarrow l i m i n f (x_{n}) = - 1 \Rightarrow {l i m i n f (x_{n})}^{3} = (- 1)^{3} = - 1 \Rightarrow l i m i n f ({x_{n}}^{3}) = {(l i m i n f x_{n})}^{3}$