applicability: $\mathbb {R}$

Definition: Convergent Real Sequence 

A convergent real sequence is a real sequence \((x_n)_{n\in\mathbb N}\), which is convergent in the metric space of real numbers \((\mathbb R,|~|)\). In other words, \((x_n)_{n\in\mathbb N}\) is convergent to the number \(x\in\mathbb R\), i.e. for each \(\epsilon > 0\) there exists an \(N\in\mathbb N\) with
\[ | x_n-x | < \epsilon\quad\quad \text{ for all }n\ge N.\]

If \((x_n)_{n\in\mathbb N}\) is convergent to the number \(x\in\mathbb R\), we write
\[\lim_{n\to\infty} x_n=x.\]

In this case, the convergent sequence is also said to be convergent to the number (or tending to the number) $x$.

| | | | | created: 2014-02-20 22:39:58 | modified: 2020-07-09 05:00:28 | by: bookofproofs | references: [581], [582]