Definition:Unbounded Divergent Sequence

Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $\mathbb{R}$.

Then $$\left \langle {x_n} \right \rangle$$ diverges to $$+\infty$$ iff $$\forall H > 0: \exists N: \forall n > N: x_n > H$$.

That is, whatever (positive) number you pick, for sufficiently large $$n$$, $$x_n$$ will exceed $$H$$.

Similarly, $$\left \langle {x_n} \right \rangle$$ diverges to $$-\infty$$ iff $$\forall H > 0: \exists N: \forall n > N: x_n < -H$$.