Real numbers/Sequence/Increasing and decreasing/Definition

Increasing sequence

A real sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ is called increasing, if ${}x_{n+1}\geq x_{n}$ holds for all ${}n\in \mathbb {N}$, and strictly increasing, if ${}x_{n+1}>x_{n}$ holds for all ${}n\in \mathbb {N}$.

A sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ is called decreasing if ${}x_{n+1}\leq x_{n}$ holds for all ${}n\in \mathbb {N}$, and strictly decreasing, if ${}x_{n+1}<x_{n}$ holds for all ${}n\in \mathbb {N}$.