Precise definition of the limit of a sequence

  • Thread starter srn
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  • #1
srn
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Main Question or Discussion Point

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].
 

Answers and Replies

  • #2
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1)They are subscripted by natural numbers in general ,i presume for simplicity and countability.
 
  • #3
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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.
 
  • #4
srn
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Great, thanks both!
 

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