Convergence Induction for Positive Sequences: Proving Limit Behavior

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Homework Help Overview

The problem involves proving the limit behavior of a sequence \( (b_n) \) converging to \( b \) under the condition that \( (a_n) \) converges to 0 and \( |b_n - b| < a_n \). The subject area pertains to sequences and limits in mathematical analysis.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the implications of the convergence of \( (a_n) \) to 0 and how it relates to the convergence of \( (b_n) \) to \( b \). There are attempts to formalize the reasoning using the epsilon-delta definition of limits.

Discussion Status

Some participants have reiterated the reasoning presented, suggesting that the argument appears valid. However, there is a request for further attempts or clarifications from others, indicating an ongoing exploration of the proof.

Contextual Notes

There is a focus on the definitions and conditions necessary for the proof, with participants examining the assumptions made regarding the sequences involved.

mathmajo
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Homework Statement


If (an) — > 0 and \bn - b\ < an, then show that (bn) — > b


Homework Equations





The Attempt at a Solution

 
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Please show your attempt at the solution.
 
Since an --> 0, given some epsilon>0, there exists and m>0 such that /an-0/=/an/<epsilon for all n>m
thus, /bn-b/</an/<epsilon for all n>m
therefore, by definition bn-->b
 
mathmajo said:
Since an --> 0, given some epsilon>0, there exists and m>0 such that /an-0/=/an/<epsilon for all n>m
thus, /bn-b/</an/<epsilon for all n>m
therefore, by definition bn-->b

Looks ok to me.
 

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