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A sequence does not converge to a

  • Thread starter eileen6a
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  • #1
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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?
 

Answers and Replies

  • #2
statdad
Homework Helper
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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].
 
  • #3
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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].


thx! related Question: Can [tex] B(x,\epsilon)[/tex] contains infinitely many[tex]x_n[/tex] in this case????
 
  • #4
statdad
Homework Helper
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"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].
 
  • #5
19
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"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].
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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