Real numbers/Sequence/Tending to infinity/Definition

Tending to ${}+\infty$

A real sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ is said to tend to ${}+\infty$, if for every ${}s\in \mathbb {R}$, there exists some ${}N\in \mathbb {N}$, such that

$x_{n}\geq s{\text{ holds for all }}n\geq N.$

The sequence is said to tend to ${}-\infty$, if for every ${}s\in \mathbb {R}$. there exists some ${}N\in \mathbb {N}$. such that

$x_{n}\leq s{\text{ holds for all }}n\geq N.$