A sequence does not converge to a

  1. a sequence [itex](x_n)[/itex] does not converge to a
    means
    infinitely many elements of [itex]\{x_n:n\in N\}[/itex] not in [itex]B(x,\epsilon)[/itex]

    why the 2 sentence equaivelent?
     
  2. jcsd
  3. statdad

    statdad 1,455
    Homework Helper

    Remember: if a sequence [tex] (x_n)[/tex] does converge to [tex] a [/tex] then, for any [tex] \espilon > 0 [/tex] there is an integer [tex] N [/tex] such that, for all
    [tex] n > N [/tex] it is true that [tex] x_n \in B(x,\epsilon)[/tex].

    With this in mind, if [tex] (x_n)[/tex] does not converge to [tex] a [/tex], it has to be true that there is no [tex] N [/tex] that satisfies the previous requirement. If saying [tex] x_n \in B(x, \epsilon)[/tex] from some point on is false, it has to be true that [tex] x_n \not \in B(x,\epsilon)[/tex] for infinitely many values of [tex] n [/tex].
     


  4. thx! related Question: Can [tex] B(x,\epsilon)[/tex] contains infinitely many[tex]x_n[/tex] in this case????
     
  5. statdad

    statdad 1,455
    Homework Helper

    "Can contains infinitely many in this case????"

    In the case of non-convergence? Sure: consider [tex] (-1)^n [/tex]. It doesn't converge
    to [tex] 1[/tex], but there are infinitely many integers (namely the even ones) for which [tex] (-1)^n \in B(1,0.1) [/tex].
     
  6. thx great example.
    how about this case?
    (x_n) converge to b.
    Can a ball centered at a contains infinitely many x_n, while a is not equal to b?
     
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