$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$

A sequence is an infinite list of numbers, for example $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\dots $, where the dots indicate that the list goes on forever. This particular sequence can be written as $1/n$ for $n\in \N$.

(Note: The set of natural numbers is denoted $\N $. In these notes, $\N =\{ 1,2,3,4, \dots \}$. Others include $0$ in $\N$. Doing so will not make anything more than a small technical difference to the theory.)

It is intuitively clear that the sequence $(1/n)$ is getting smaller and is heading towards $0$. We call $0$ the limit of the sequence. This intuitive definition of limit is easy to understand but the precise mathematical definition is harder to grasp immediately. We shall define sequences and the mathematical definition of limit. We focus mainly on examples to begin with.

Definition
A sequence is an infinite list of numbers $a_1$, $a_2$, $a_3$, $\dots$. We shall write $(a_n)$ or simply $a_n$, where $n\in \N$.

Examples
The following are all examples of sequences.

1. $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\dots$
2. $0$, $0$, $0$, $0$, $0$, $\dots $
3. $1$, $0$, $\frac{1}{2}$, $0$, $\frac{1}{3}$, $0$, $\frac{1}{4}$, $0$,$\dots$
4. $1$, $2$, $3$, $4$, $\dots$
5. $1$, $-1$, $1$, $-1$, $1$, $\dots$
6. $n^2$, $n\in \N$,
7. $2n$, $n\in \N$, is the sequence of even numbers,
8. $2n-1$, $n\in \N$, is the sequence of odd numbers,
9. $\cos(n) $, $n\in \N $,
10. $\displaystyle \frac{3n+\sin n}{(2n+7)(n^2+1)}$, $n\in \N$,
11. $(-1)^{n+1}$ and $\cos \left((n-1)\pi \right)$, $n\in \N$, give the same sequence. (In fact, the same sequence as in example (5).

Remark
We can define a sequence as a function from $\N $ to $\R$. That is, $f:\N \to \R$ is defined by $f(n)=a_n$. This is just an alternative way of viewing the concept of sequences, it is equally as valid as our notion of viewing it as a list.