Purchase Solution

Sequences Limit Consequences

Not what you're looking for?

Ask Custom Question

(a) Prove this operation:
Let {xn} and {yn} be convergent sequences.
The sequence{zn} where zn:=xn-yn converges and lim (xn-yn)=lim zn=lim xn-limyn

What I attempted was this:

Suppose {xn} and {yn} are convergent sequences and write zn:=xn-yn. Let x:=lim xn, y:=lim yn and z:=x-y
Let epsilon>0 be given. Find M1 s.t. for all n =>M1 we have |xn-x|<epsilon/2. Find M2 s.t. for n => M2 we have |yn-y|<epsilon/2
Take M:=max{M1,M2} For all n => m We have, |zn-z|=|(xn-yn)-(x-y)|

This is the part I get stuck at. I think I want to ultimately show the |zn-z|< epsilon, but I don't know if I am pursuing this correctly.

(b) Let xn:=[(n-1)(-1)^n]/n, find the lim sup xn and lim inf xn.

I tried doing the squeeze lemma, but that didn't work.

Thank you.

Purchase this Solution
Solution Summary

The expert proves that the subtraction of two convergent sequences is also convergent. Also, determining the liminf and limsup for a given sequence. Squeeze lemmas are examined.

Solution Preview

Hello and thank you for posting your question to BrainMass.
The solution is ...

Solution provided by:
Mert Mustafa, PGDip
Education
  • PGDip , University of Kent
  • BSc, University of Kent
Recent Feedback
  • "Thank you so much!"
  • "Thank you so much!"
  • "Thank you very much."
  • "Thank you so much. As I read the pages, I realized that you deserve way more than 7 credits for this! This was truly helpful. "
  • "Thank you very much. I always appreciate the amount of explanatory detail you provide in your answers! "
Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

View More Free Quizzes
Related BrainMass Solutions

  • Sequences and convergence

    The solution looks at sequences and shows that various sequences are monotone sequences. It also shows how to illustrate the fact that two sequences converge to the same limit.

  • Sequences and limits

    45227 Sequences and limit boundaries 1) If (bn) is a bounded sequence and lim(an) = 0, show that the lim(anbn) = 0. Explain why Theorem 3.2.3 cannot be used.

  • Real Analysis : Sequences and Limits

    27194 Real Analysis : Sequences and Limits Show that if f be a function defined on A, and c be a limit point of A.

  • What is a limit?

    By taking the limit, we can see clearly the tendency of some sequence such as {1+1/n} as n goes to infinity. However, not all sequences have limit,which you will learn later in college.

  • Sequences & series

    x if x>0, test the series for convergance. 7.Using convergence tests (limit comparison test, ratio test, root test) test whether the series is convergent or divergent. The explanations are in the attached file. Sequences and series are evaluated.

  • Analysis of Limit Problem : Sequences of Positive Numbers

    108396 Analysis of Limit Problem : Sequences of Positive Numbers Please see the attached file for the fully formatted problem. Please see the attached file for the complete solution. Thanks for using BrainMass.

  • 40 Problems : Sequences, Series, Convergence, Divergence and Limits

    Determine the limit(where there is one ) for the sequences whose nth terms are given by the following formulae. Give brief justification.

  • Real Analysis : Convergent Sequences and Limits

    34399 Real Analysis : Convergent Sequences and Limits Prove the sequence defined by a1 = 0, a2=1 and a(n+2)= (a(n+1)+a(n))/2 for n>=0, converges and find the limit.

  • Converging and diverging infinite sequences and limits.

    75425 Converging and diverging infinite sequences and limits. 1) Does this sequence converge or diverge? Please show explanation/proof. 2) If the series converges, what is its limit? Please show work.

View More Related Solutions