Precise definition of the limit of a sequence

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Discussion Overview

The discussion revolves around the precise definition of the limit of a sequence, focusing on the conditions involving natural numbers and the epsilon-delta criterion. It explores theoretical aspects of mathematical definitions.

Discussion Character

  • Technical explanation

Main Points Raised

  • One participant questions the necessity of using natural numbers (n_0 ∈ N) in the definition, suggesting that real numbers (n_0 ∈ R) could also suffice.
  • Another participant proposes that the use of natural numbers is likely for simplicity and countability.
  • There is a discussion about whether the condition |x_n - a| < ε is essential, with one participant arguing that |x_n - a| ≤ ε could also be acceptable, as both conditions must hold for all ε > 0.
  • Another participant agrees that both definitions (using <ε or ≤ε) are equivalent.

Areas of Agreement / Disagreement

Participants generally agree that using n_0 ∈ R is a valid approach and that both forms of the epsilon condition are equivalent. However, the discussion does not reach a consensus on the necessity of the original definitions.

Contextual Notes

The discussion highlights assumptions about the definitions of limits and the implications of using different sets of numbers, but does not resolve the nuances of these definitions.

srn
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In the definition,

1) why must you find a n_0 \in N such that \forall N \geq n_0? You might as well say find a n_0 \in R such that \forall N &gt; n_0. Just a matter of simplicity?

2) Why must |x_n - a| &lt; \epsilon hold? I think |x_n - a| \leq \epsilon is fine as well, given that it must hold \forall \epsilon &gt; 0.
 
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1)They are subscripted by natural numbers in general ,i presume for simplicity and countability.
 
1) Yes, taking n_0\in \mathbb{R} works as well. But it is often simper to take n_0\in \mathbb{N}.

2) Having <ε or ≤ε makes no difference. Both definitions work and are equivalent.
 
Great, thanks both!
 

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