My task is to write a precise mathematical statement that "the sequence $(a_n)$ does not converge to a number $\mathscr l$"
So, I have my definition of a convergent sequence: "$\forall\varepsilon>0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr l|<\varepsilon$ $\forall n \in \Bbb N$ with $n>N$"
Would the correct negation of this be "$\forall\varepsilon>0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr l|>\varepsilon$ $\forall n \in \Bbb N$ with $n>N$"?
It doesn't seem that this is the answer as the next part of my task is to prove that a sequence is divergent using my formed proof, but it'd be difficult to do since it's a general proof of divergence and not just a proof that $(a_n)$ doesn't converge a specific number $\mathscr l$
Perhaps I should find a prove that $(a_n)$ tends to $\pm\infty$? This is more simple but it does not include monotone sequences such as $x_n:=(-1)^n$.
Can someone assist me with this task? All comments and answers are appreciated.