Explanation of Cauchy Proof for Convergent Sequence

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    Cauchy Proof
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Discussion Overview

The discussion centers around the proof of the Cauchy criterion for convergent sequences, specifically focusing on the interpretation of the interval length derived from the convergence definition. Participants explore the mathematical reasoning behind the length of the interval defined by the convergence of a sequence to a limit.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant outlines that for a convergent sequence { a_n } converging to A, there exists a positive integer N such that for all n, m >= N, the terms a_n and a_m lie within the interval (A - e, A + e).
  • Another participant questions the derivation of the interval length 2e, seeking clarification on how this length is calculated.
  • Several participants confirm that the length of the interval (A - e, A + e) is indeed 2e, explaining it as the difference (A + e) - (A - e).
  • One participant expresses a feeling of inadequacy after realizing the simplicity of the explanation provided.

Areas of Agreement / Disagreement

Participants generally agree on the calculation of the interval length as 2e, but there is a moment of uncertainty expressed by one participant regarding the initial explanation.

Contextual Notes

The discussion does not resolve the initial confusion about the proof's presentation, as it relies on the understanding of interval lengths and the properties of convergent sequences.

lion0001
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Suppose { a_n } converges to A. Choose e > 0, THere is a positive integer N such that, if
n, m >= N , then A - e < a_n < A + e and A - e < a_m < A + e
Thus for all n, m >= N we find a_n ∈ ( A - e , A + e ) and
a_m ∈ ( A - e , A +e ) . the set ( A - e, A +e ) is an interval of length 2e , hence the difference between a_n, and a_m is less then 2e

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i don't understand how they get 2e , ( a set of length 2e ?? )
 
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what is (A + e) - (A - e)
 
It's simply (A + e) - (A - e) = 2e.
Same as I would say the interval (2,5) has length 5-2=3.
 
nicksauce said:
It's simply (A + e) - (A - e) = 2e.
Same as I would say the interval (2,5) has length 5-2=3.

Excellent answer, OMG this proved that i am an idiot =(
 

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