TITLE: Exact definition of convergence
QUESTION [6 upvotes]: 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?

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 $M\gt n$, $|x_M-L|\lt\epsilon$. 
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?

REPLY [7 votes]: 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.