New Question: Proving Divergence of a Sequence After I negated the definition of a convergent sequence, I ended up with the following mathematical statement: 

$$\exists\ \epsilon &gt; 0,\ \forall\ N \in \mathbb R\ \exists\ \mathbb N \ni n &gt; N : |x_n - l| \ge \epsilon$$

Is this correct? I'd like clarification.

Anyway, I'm now asked to use my negation of the definition of convergence to prove that $(a_n)=((-1)^nn)$ is divergent. Can someone show me how?
Your negation is right, except for the little mistake that $N\in \mathbb{N}$. For your example, take $\varepsilon=1$. $\forall N \in \mathbb{N}$, take $n=\max\{2N,2l\}$. So we have examples that your statement is satisfied $\forall l\in \mathbb{R}$.
Your negation is right, except for the little mistake that $N\in \mathbb{N}$. For your example, take $\varepsilon=1$. $\forall N \in \mathbb{N}$, take $n=\max\{2N,2l\}$. So we have examples that your statement is satisfied $\forall l\in \mathbb{R}$.