Q: Sequence definition of non-convergence (Not diverging definition)

Question

I have that convergence requires that the following be true

$$ \forall\epsilon>0\exists N: n\ge N:|a_n-a|<\epsilon $$

for some sequence $\{a_n\}_{n\in\mathbb{N}}$ that converges to $a$ for $n\to\infty$.

I want to use the reverse of this requirement. I do not know how to express it but I am guessing

$$ \exists\epsilon>0\forall N:n\ge N:|a_n-a|\ge\epsilon $$

it is expressed like the above. Can you help me out?

Answer 1

A somewhat cleaner definition of the limit of a sequence is: $$ \forall \epsilon>0, \exists N\in \mathbb{N}, \forall n \geq N, |a_n-a|<\epsilon $$ Then the opposite is: $$ \exists \epsilon>0, \forall N\in \mathbb{N}, \exists n \geq N, |a_n-a|\geq\epsilon $$