Definition of a Divergent Sequence

I have that the definition of a convergent sequence is:

($\exists$ L $\in$ $\mathbb{R}$)($\forall$ $\epsilon$ > 0)($\exists$ N $\in$ $\mathbb{N}$)($\forall$ n $\in$ $\mathbb{N}$)[n $\ge$ N $\Rightarrow$ |$x_n-L|$ < $\epsilon$]

Would it follow that the definition of divergence is the negation:

($\forall$ L $\in$ $\mathbb{R}$)($\exists$ $\epsilon$ > 0)($\forall$ N $\in$ $\mathbb{N}$)($\exists$ n $\in$ $\mathbb{N}$)[n $\ge$ N and |$x_n-L|$ $\ge$ $\epsilon$]

or would it simply be the definition of convergence except making it so there does not exist L in the reals so:

(∄ L $\in$ $\mathbb{R}$)($\forall$ $\epsilon$ > 0)($\exists$ N $\in$ $\mathbb{N}$)($\forall$ n $\in$ $\mathbb{N}$)[n $\ge$ N $\Rightarrow$ |$x_n-L|$ < $\epsilon$]

1 Answer

Your two mathematical sentences are equivalent, so it doesn't matter. (There is a typo in the English version of the second sentence: you meant: "there does not exist".)