- #1

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means

infinitely many elements of [itex]\{x_n:n\in N\}[/itex] not in [itex]B(x,\epsilon)[/itex]

why the 2 sentence equaivelent?

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- Thread starter eileen6a
- Start date

- #1

- 19

- 0

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

statdad

Homework Helper

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[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]

- #3

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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 thatthere 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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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

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- 0

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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