Definition:Unbounded Divergent Sequence/Real Sequence

Definition
Let $\sequence {x_n}$ be a sequence in $\R$.

Divergence to positive infinity
Then $\sequence {x_n} $ tends to $+\infty$ or diverges to $+\infty$ :
 * $\forall H > 0: \exists N: \forall n > N: x_n > H$.

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

Divergence to negative infinity
Similarly, $\sequence {x_n} $ tends to $-\infty$ or diverges to $-\infty$ :
 * $\forall H > 0: \exists N: \forall n > N: x_n < -H$.

We write:
 * $x_n \to +\infty$ as $n \to \infty$

or:
 * $x_n \to -\infty$ as $n \to \infty$.

Divergence to infinity
$\sequence {x_n}$ tends to $\infty$ or diverges to $\infty$ :
 * $\forall H > 0: \exists N: \forall n > N: \size {x_n} > H$

Also see

 * Definition:Divergent Complex Sequence to Infinity
 * Definition:Divergent Sequence
 * Definition:Infinite Limit at Infinity