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Definition of Limit

Limit of a sequence
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.



Definition of Sequence

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



The Squeeze Rule

The squeeze rule is one of the most useful tools in the study of sequences. It allows us to show that a sequence converges to a limit if we ‘sandwich’ it between two other sequences that converge to the same limit. In effect the sequence gets squeezed between the other two.

Theorem: The Squeeze Rule

Suppose that a_n, b_n and c_n are sequences such that a_n\leq b_n \leq c_n for all n. If a_n\to l and c_n \to l, then b_n \to l.

Proof.

Let x_n=b_n-a_n and y_n=c_n-a_n. Then \displaystyle y_n = c_n-a_n \to  l - l =0 .
So as y_n \to 0 , given any \varepsilon >0 there exists N such that |y_n-0| <\varepsilon for all n\geq N. But for n\geq N and as 0\leq x_n \leq y_n,

|x_n-0|=|x_n|\leq |y_n| = |y_n-0|<\varepsilon .

Therefore, x_n\to 0 and so \displaystyle b_n = x_n+a_n \to 0+ l =l. \hfil \square .