The intuitive concept of the limit of a sequence is very simple. The terms in the sequence $\frac{1}{n}$ get closer to $0$ as $n$ gets larger. However the words `get closer’ is not very mathematical and in fact is misleading. (The sequence $0,0,0,0,\dots $ has limit $0$. In what sense would the terms of the sequence be getting closer to $0$?)

The definition of limit took mathematicians many years to be arrived — it took many years for them to realise that a good definition was needed — and so it is not surprising that it is one of the hardest concepts to understand in mathematics.

**Definition**

The number $a$ is a **limit** of the sequence $(a_n)$ if for all $\varepsilon >0$, there exists $N\in \mathbb{N}$ such that for all $n> N$ we have $|a_n-a|<\varepsilon $.
We write $a_n\to a$ or $\displaystyle \lim_{n\to \infty } a_n$ and say `$a_n$ **tends** to $a$, or $(a_n)$ **converges** to $a$, as $n$ tends to infinity’.

If a sequence $(a_n)$ has a limit, then we say that $(a_n)$ is **convergent**.

**Remark**

Some authors use $n\geq N$ rather than $n>N$. It makes no difference to the theory.

The above definition of limit is certainly far from intuitive but understanding it is vital for the rest of the book. Let’s do some examples.

**Example**

The limit of $a_n = 1/n$ is $0$.

**Proof**

Let $N$ be a natural number greater than $1/\varepsilon $. Then, $1/N< \varepsilon $ and we have
\begin{align*}
|a_n - a | &= | 1/n - 0 | \\
&= |1/n| \\
&= 1/n \\
&\leq 1/N , {\text{ for }} n\geq N, \\
&< \varepsilon .
\end{align*}
In this example, note that it is not obvious why we should choose $N$ to be greater than $1/\varepsilon$. This is a common point of confusion and we will come back to it later.

**Remark**

The type of proof used above to show that $1/n\to 0$ is called an $\varepsilon-N$ proof.

A

**Examples**

The following are all examples of sequences.

- $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\dots$
- $0$, $0$, $0$, $0$, $0$, $\dots $
- $1$, $0$, $\frac{1}{2}$, $0$, $\frac{1}{3}$, $0$, $\frac{1}{4}$, $0$,$\dots$
- $1$, $2$, $3$, $4$, $\dots$
- $1$, $-1$, $1$, $-1$, $1$, $\dots$
- $n^2$, $n\in \N$,
- $2n$, $n\in \N$, is the sequence of even numbers,
- $2n-1$, $n\in \N$, is the sequence of odd numbers,
- $\cos(n) $, $n\in \N $,
- $\displaystyle \frac{3n+\sin n}{(2n+7)(n^2+1)}$, $n\in \N$,
- $(-1)^{n+1}$ 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.

Suppose that and are sequences such that for all . If and , then .

Let and . Then

So as , given any there exists such that for all . But for and as ,

Therefore, and so ]]>