Real numbers/Sequence/Limit and convergence/Definition

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ denote a real sequence, and let ${\displaystyle {}x\in \mathbb {R} }$. We say that the sequence converges to ${\displaystyle {}x}$, if the following property holds.

For every positive ${\displaystyle {}\epsilon >0}$, ${\displaystyle {}\epsilon \in \mathbb {R} }$, there exists some ${\displaystyle {}n_{0}\in \mathbb {N} }$, such that for all ${\displaystyle {}n\geq n_{0}}$, the estimate

${\displaystyle {}\vert {x_{n}-x}\vert \leq \epsilon \,}$

holds.

If this condition is fulfilled, then ${\displaystyle {}x}$ is called the limit of the sequence. For this we write

${\displaystyle {}\lim _{n\rightarrow \infty }x_{n}:=x\,.}$

If the sequence converges to a limit, we just say that the sequence converges, otherwise, that the sequence diverges.