2.3 Sequences of real numbers: basic properties

Sequences can behave in surprisingly different ways, but some have properties that we will be able to use to analyse their behaviour. We begin by looking at monotone and bounded sequences.

Monotone sequences

We first consider sequences whose terms grow ever larger, or ever smaller.

Definition 2.10.

Let $(a_{n})_{n\in\mathbb{N}}$ be a sequence.

1

We say $(a_{n})_{n\in\mathbb{N}}$ is increasing if $a_{n}<a_{n+1}$ for all $n\in\mathbb{N}$ ;
2

We say $(a_{n})_{n\in\mathbb{N}}$ is decreasing if $a_{n}>a_{n+1}$ for all $n\in\mathbb{N}$ .

In other words, $(a_{n})_{n\in\mathbb{N}}$ is increasing if $a_{1}<a_{2}<a_{3}<\cdots$ and is decreasing if $a_{1}>a_{2}>a_{3}>\cdots$ .

Example 2.11.

For the sequences from Example 2.2, it is clear that

•

$(1/n)_{n\in\mathbb{N}}$ is decreasing,
•

$(n)_{n\in\mathbb{N}}$ is increasing and
•

$((-1)^{n})_{n\in\mathbb{N}}$ is neither increasing nor decreasing.

Sometimes we say strictly increasing or strictly decreasing, to emphasise that the inequalities in Definition 2.10 are strict (as opposed to $\leq$ or $\geq$ ). We also have terms for the non-strict version:

Definition 2.12.

Let $(a_{n})_{n\in\mathbb{N}}$ be a sequence.

1

We say $(a_{n})_{n\in\mathbb{N}}$ is non-decreasing if $a_{n}\leq a_{n+1}$ for all $n\in\mathbb{N}$ ;
2

We say $(a_{n})_{n\in\mathbb{N}}$ is non-increasing if $a_{n}\geq a_{n+1}$ for all $n\in\mathbb{N}$ ;
3

We say $(a_{n})_{n\in\mathbb{N}}$ is monotone if it is either non-decreasing or non-increasing.

Note that any increasing (respectively, decreasing) sequence is automatically non-decreasing (respectively, non-increasing) and therefore monotone. More succinctly,

Increasing	$\Longrightarrow$	Non-decreasing	$\Longrightarrow$
				Monotone.
Decreasing	$\Longrightarrow$	Non-increasing	$\Longrightarrow$

Exercise 2.13.

Determine whether the following sequences are increasing, non-decreasing, decreasing, non-increasing or monotone.

(i)

$(2,3,5,7,11,13,17,19,23,\dots)$ .
(ii)

$(1,1,2,1,2,3,1,2,3,4,\dots)$ ;
(iii)

$(5,5,5,5,\dots)$ ;
(iv)

$(0,1/2,2/3,3/4,4/5,\dots)$ ;
(v)

$(1,2,2,3,3,3,4,4,4,4,\dots)$ ;

Complete each cell of the table with a or to record your answers.

	Increasing	Non-decreasing	Decreasing	Non-increasing	Monotone
(i)
(ii)
(iii)
(iv)
(v)

Exercise 2.14.

Suppose $(a_{n})_{n\in\mathbb{N}}$ is a non-increasing sequence. Show that $(-a_{n})_{n\in\mathbb{N}}$ is a non-decreasing sequence.

Bounded and unbounded sequences

The sequences $(n)_{n\in\mathbb{N}}$ and $(1-1/n)_{n\in\mathbb{N}}$ are both increasing. However, we can see from their graphs (shown in Figure 2.4) that they exhibit very different behaviours: the first sequence increases without bound, while the second sequence never exceeds 1.

(n)_{n\in\mathbb{N}}

(1-1/n)_{n\in\mathbb{N}}

Figure 2.4: Graphs of the increasing sequences

(n)_{n\in\mathbb{N}}

and

(1-1/n)_{n\in\mathbb{N}}

Definition 2.15.

Let $(a_{n})_{n\in\mathbb{N}}$ be a sequence.

1

We say $(a_{n})_{n\in\mathbb{N}}$ is bounded above if there exists some $M\in\mathbb{R}$ such that $a_{n}\leq M$ for all $n\in\mathbb{N}$ ;
2

We say $(a_{n})_{n\in\mathbb{N}}$ is bounded below if there exists some $L\in\mathbb{R}$ such that $a_{n}\geq L$ for all $n\in\mathbb{N}$ ;
3

We say $(a_{n})_{n\in\mathbb{N}}$ is bounded if it is both bounded above and bounded below. Otherwise, we say $(a_{n})_{n\in\mathbb{N}}$ is unbounded.

Remark 2.16.

Note that, by definition, the sequence $(a_{n})_{n\in\mathbb{N}}$ is bounded above if and only if the set $\{a_{n}:n\in\mathbb{N}\}$ is bounded above in the sense introduced in Definition 1.1.

Example 2.17.

Consider the three sequences from Example 2.2.

1.

Since $0<1/n\leq 1$ for all $n\in\mathbb{N}$ , the sequence $(1/n)_{n\in\mathbb{N}}$ is bounded. Similarly, the increasing sequence $(1-1/n)_{n\in\mathbb{N}}$ is also bounded.
2.

Since $n\geq 0$ for all $n\in\mathbb{N}$ , the sequence $(n)_{n\in\mathbb{N}}$ is bounded below. However, given any $M\in\mathbb{R}$ , we can find some $n\in\mathbb{N}$ such that $n>M$ . This tells us that $(n)_{n\in\mathbb{N}}$ is not bounded above and is therefore unbounded.
3.

Since $-1\leq(-1)^{n}\leq 1$ for all $n\in\mathbb{N}$ the sequence $((-1)^{n})_{n\in\mathbb{N}}$ is bounded.

We can interpret Definition 2.15 in terms of our visual aids. For instance, a sequence is bounded above if and only if there exists some $M\in\mathbb{R}$ such that the graph of the sequence lies entirely below the horizontal line $y=M$ . See Figure 2.5.

(a) A bounded sequence: the terms lie in a horizontal ‘corridor’ between

y=L

and

y=M

(b) The sequence

(n^{2})_{n\in\mathbb{N}}

is unbounded. For any threshold

M\in\mathbb{R}

, there exists a term in the sequence which lies above the line

y=M

Figure 2.5: Bounded and unbounded sequences.

Exercise 2.18.

Determine whether the following sequences are bounded above

(i)

$\displaystyle\Big{(}\frac{n+2}{2n^{2}-n}\Big{)}_{n\in\mathbb{N}}$ ;
(ii)

$\displaystyle\Big{(}\frac{10n^{4}+n^{2}+4}{6n^{3}-4n^{2}}\Big{)}_{n\in\mathbb{% N}}$ .

A video walk-through of a solution to part (i) can be found here.

Example 2.19.

Consider the sequence $(2^{n})_{n\in\mathbb{N}}$ . Using the binomial theorem we see that

2^{n}=(1+1)^{n}=\sum_{k=0}^{n}\binom{n}{k}=1+n+\sum_{k=2}^{n}\binom{n}{k}\geq 1% +n.

This is a particular instance of the Bernoulli inequality from IMU (see also Worksheet 1). Given any $M\in\mathbb{R}$ , we can find some $n\in\mathbb{N}$ such that $n>M$ . Thus, $2^{n}\geq n+1>M+1>M$ . This tells us that $(2^{n})_{n\in\mathbb{N}}$ is not bounded above and is therefore unbounded.

Exercise 2.20.

Show that a sequence $(a_{n})_{n\in\mathbb{N}}$ is bounded if and only if there exists some $R>0$ such that $|a_{n}|\leq R$ for all $n\in\mathbb{N}$ .

Exercise 2.21.

Suppose $(a_{n})_{n\in\mathbb{N}}$ is bounded below. Show that $(-a_{n})_{n\in\mathbb{N}}$ is bounded above.