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In $\beta\mathbb N$ there are no non-trivial convergent sequences. I want to show this, but what is the meaning of non -trivial convergent sequence?

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Any sequence which is eventually constant converges (in any topology). These might be called the "trivial convergent sequences."

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could you give me example for non-trivial convergent squence? and trivial convergence? –  ege Nov 18 '12 at 7:09
    
Do you mean unique limit for sequence? –  ege Nov 18 '12 at 7:11
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@ege: No, Qiaochu is not talking about unique limits. The sequence $\left\langle\frac1n:n\in\Bbb Z^+\right\rangle$ is an example of a non-trivial convergent sequence in the usual topology on $\Bbb R$. The sequence $\langle x_n:n\in\Bbb N\rangle$ such that $x_n=n$ if $n<100$ and $x_n=-1$ if $n\ge 100$ is a trivial convergent sequence: it’s eventually constant at $-1$, so of course it has to converge to $-1$ no matter what topology on $\Bbb R$ is considered. –  Brian M. Scott Nov 18 '12 at 7:27
    
Oh, yes, I can see. Thanks so much . –  ege Nov 18 '12 at 7:50

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