Definition of a Divergent Sequence
$\Begingroup$ I Have That the Definition of a Convergent Sequence Is: ($\Exists$ L $\In$ $\Mathbb{R}$)($\Forall$ $\Epsilon$ > 0)($\Exists$ N $\In$...
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".)
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