Real numbers/Sequence/Limit and convergence/Definition

Convergent sequence

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

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

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

holds.

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

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

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