Proving Existence of Limit of Sequence {xn}

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Homework Help Overview

The discussion revolves around proving the existence of the limit of a sequence {xn} that satisfies the condition 0 ≤ x_{m+n} ≤ x_{m} + x_{n}. Participants are exploring the implications of this condition on the limit as n approaches infinity.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Some participants are attempting to derive relationships between the terms of the sequence, such as expressing x2 and x3 in terms of x1. Others are questioning how these relationships might lead to insights about the limit.

Discussion Status

Participants are actively engaging with the problem, expressing confusion and seeking clarification. There is an indication that some guidance has been offered regarding the form of xn, but no consensus or resolution has been reached.

Contextual Notes

Some participants express difficulty in understanding the implications of the given condition on the sequence, indicating a need for further exploration of the assumptions involved.

cristina89
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Be {xn} a sequence that satisfies the condition 0 ≤ [itex]x_{m+n}[/itex] ≤ [itex]x_{m}[/itex] + [itex]x_{n}[/itex]. Prove that [itex]lim_{n ->∞}[/itex] xn/n exists.

I'm kind of lost in this.
 
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cristina89 said:
Be {xn} a sequence that satisfies the condition 0 ≤ [itex]x_{m+n}[/itex] ≤ [itex]x_{m}[/itex] + [itex]x_{n}[/itex]. Prove that [itex]lim_{n ->∞}[/itex] xn/n exists.

I'm kind of lost in this.

I would start by thinking like this:

x2<=x1+x1

x3<=x2+x1<=x1+x1+x1

etc. What can you make of that?
 
Ughh I'm still lost :S
 
Hi cristina89! :smile:

Can you write xn in the form that Dick suggested?
 

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