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Precise definition of the limit of a sequence

  1. Aug 18, 2012 #1

    srn

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    In the definition,

    1) why must you find a [itex]n_0 \in N[/itex] such that [itex]\forall N \geq n_0[/itex]? You might as well say find a [itex]n_0 \in R[/itex] such that [itex]\forall N > n_0[/itex]. Just a matter of simplicity?

    2) Why must [itex]|x_n - a| < \epsilon[/itex] hold? I think [itex]|x_n - a| \leq \epsilon[/itex] is fine as well, given that it must hold [itex]\forall \epsilon > 0[/itex].
     
  2. jcsd
  3. Aug 18, 2012 #2
    1)They are subscripted by natural numbers in general ,i presume for simplicity and countability.
     
  4. Aug 18, 2012 #3

    micromass

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    Staff Emeritus
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    1) Yes, taking [itex]n_0\in \mathbb{R}[/itex] works as well. But it is often simper to take [itex]n_0\in \mathbb{N}[/itex].

    2) Having <ε or ≤ε makes no difference. Both definitions work and are equivalent.
     
  5. Aug 18, 2012 #4

    srn

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    Great, thanks both!
     
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