Proving Converging Sequences: {an}, {an + bn}, {bn}

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Homework Help Overview

The discussion revolves around the convergence of sequences, specifically addressing the claim that if the sequences {an} and {an + bn} are convergent, then the sequence {bn} must also be convergent. Participants are exploring the implications of this statement using delta-epsilon definitions.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to prove the statement using delta-epsilon definitions and are questioning whether their approach is correct. There is discussion about deriving relationships between the limits of the sequences and expressing bn in terms of an and an + bn.

Discussion Status

Some participants have provided hints and affirmations regarding the approach, while others express confusion about the next steps in the proof. The discussion is ongoing, with various interpretations and methods being explored.

Contextual Notes

Participants are working under the constraints of formal definitions of convergence and are seeking to clarify the relationships between the sequences involved. There is an emphasis on proving the statement rather than providing counterexamples.

Nan1teZ
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Homework Statement



Prove or give a counterexample: If {an} and {an + bn} are convergent sequences, then {bn} is a convergent sequence.


2. The attempt at a solution

Ok I couldn't think of any counterexamples, so I tried to prove it using delta epsilon definitions:

|an - L| < E
|an + bn - M| < E
want to show: |bn - N| < E

Is this the right approach?
 
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Yes, and M=L+N, right? You can certainly prove that. It's the correct approach.
 
yeah I got the N = M-L part. But then after that I go in circles trying to show it is < Epsilon. =[

What's the little trick?
 
Hint: [tex]\lim_{n \to \infty} \left\{ a_{n} + b_{n} \right\} = \lim_{n \to \infty} a_{n} + \lim_{n \to \infty} b_{n}[/tex]
 
And bn=(an+bn)+(-an).
 

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