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Let us consider a sequence $x_n$. Now let it converge to a limit $L$. Now which one of the following is the correct definition of convergence?

  1. A sequence $x_n$ is said to be convergent to a limit $L$ if given any integer $n$ there exists a positive real number $\epsilon$ such that for all $M\gt n$, $|x_M-L|\lt\epsilon$.

  2. A sequence $x_n$ is said to be convergent to a limit $L$ if given any real positive number $\epsilon$ there exists an integer $n$ such that for all $M\gt n$, $|x_M-L|\lt\epsilon$.

If the two definitions are equivalent then how to prove it?

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You overuse commas a little bit... –  Arturo Magidin Nov 27 '11 at 18:43
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1 Answer

up vote 4 down vote accepted

The second definition is correct; the first definition is incorrect.

For example, the sequence $x_n = (-1)^n$ satisfies your first definition with both $L=1$ and $L=-1$: given any $n\gt 0$, let $\epsilon=3$. Then for every $M\gt n$ we have $|x_M-1|\lt 3$ and $|x_M+1|\lt 3$.

In fact, your first definition is satisfied by any bounded sequence (in particular, any convergent sequence) with any value of $L$. If you suspect that convergent sequences should have only one limit, that should be tip-off that the first definition is incorrect.

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Thank you, sir. –  user16186 Nov 28 '11 at 3:06
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