Real Sequences

Definition: Real Sequence

A real sequence or sequence of real numbers is a real-valued function $a:\mathcal{D}\subseteq {\mathbb{N}}_0\to \mathbb{R}$ whose domain $\mathcal{D}$ is either ${\mathbb{N}}_0$ or a finite subset of ${\mathbb{N}}_0$ which has the form $\mathcal{D}=\{0,1,2,3,\dots ,n\},n\in \mathbb{N}$.

NOTATION

We usually denote $a$ by $(a_n{)}_{n\in \mathcal{D}}$ or $\{a_n{\}}_{n\in \mathcal{D}}$. If $\mathcal{D}$ is not really relevant, we can drop it and write just $(a_n)$ or $\{a_n\}$. If $\mathcal{D}={\mathbb{N}}_0$ or $\mathcal{D}=\mathbb{N}$, we write $\{a_n{\}}_{n=0}^{\infty }$ and $\{a_n{\}}_{n=1}^{\infty }$, respectively.

Instead of $a(0),a(1),a(2),\dots ,a(k)$, we write $a_0,a_1,a_2,\dots ,a_k$.