QuestionQuestion

Transcribed TextTranscribed Text

Let on be a bounded sequence. In this workshop, we will derive several results about the numbers called lim inf an and lim supan- Let. us define new sequences bn and Cn as follows: In other words, bn is the greatest lower bound of the set {ax : kej and le n} and Gn is the least upper bound of the same set {at : * € J and le n). Problem (1) Prove that on is a bounded monotone sequence. This implies that bn converges to some real number. This real number is called lim inf an. In other words, ba converges to liminfan- Problem (2) Prove that Cn is a bounded monotone sequence. This implies that Cn converges to some real number. This real number is called limsupan- In other words, Cn converges to lim sup an- Problems (1) and (2) give us the definitions of lim inf an and lim supon- Now we can prove some theorems about these concepts: Problem (3) Prove the existence of a suhsequence of an such that this subsequence converges to lim inf an- Problem (4) Prove the existence of a subsequence of an such that this subsequence converges to lim sup on- Problem (5) Prove: If an has a subsequence that converges to a, then lim inf on <05 limsupún- Problem (6) Prove: If liminfan = limsupan then an converges to a, where a = lim inf an = lim supan- Problem (7) Prove: If an converges to some a, then lim inf an = lim sup On. The result in Problem (6) then implies a = liminfan = lim sup on-

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.pdf.

    $40.00
    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Real Analysis Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats