---
tag: 0CTV783HYT
kind: definition
name: Real sequence
msc: [40A05]
defines: 04CC6K5VDT
uses: [0QE8STGJH2]
sources:
  - url: https://en.wikipedia.org/wiki/Sequence
    locator: Limits and convergence
---

## Definition

A real sequence is a [[sequence|0QE8STGJH2]] in the set $\mathbb{R}$ of real numbers.
Thus it is a function $a \colon \mathbb{N} \to \mathbb{R}$, and each term $a_n$ is a real number.

## Remarks

Examples are $a_n = 1/(n + 1)$, $a_n = (-1)^n$ and $a_n = 2^n$.
The real sequences form a real vector space under the termwise operations $(a_n) + (b_n) = (a_n + b_n)$ and $c\,(a_n) = (c\,a_n)$.
The [[limit of a sequence|0XKWSSY8P2]] is defined first for real sequences in most courses.

## Explanations

### L1 human

A real sequence is an endless list of real numbers $a_0, a_1, a_2, \dots$.
You can draw it as points $(n, a_n)$ in the plane, one point for each natural number $n$.