# Real Sequences

## Sequence of real numbers

A sequence of real numbers (or a real sequence) is defined as a function $f: \mathbb{N} \to \mathbb{R}$ , where $\mathbb{N}$ is the set of natural numbers and $\mathbb{R}$ is the set of real numbers. Thus, $f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a function which produces a sequence of real numbers $r_n$ . It’s customary to write a sequence as form of functions in brackets, e.g.; $\langle f(n) \rangle$ , ${ f(n) }$ . We can alternatively represent a sequence as the function with natural numbers as subscripts, e.g., $\langle f_n \rangle$ , ${ f_n }$ . This alternate method is a better representation of a sequence as it distinguishes ‘a sequence’ from ‘a function’. We shall use $\langle f_n \rangle$ notation and when writen $\langle f_n \rangle$ , we mean $\langle f_1, f_2, f_3, \ldots, f_n, \ldots \rangle$ a sequence with infinitely many terms. Since all of ${ f_1, f_2, f_3, \ldots, f_n, \ldots }$ are real numbers, this kind of sequence is called a sequence of real numbers.

## Examples of Sequences

1. Like $f(x)=\dfrac{1}{x} \forall x \in \mathbb{R}$ is a real-valued-function, $f(n)=\dfrac{1}{n} \forall n \in \mathbb{N}$ is a real sequence.

Putting consecutive values of $n \in \mathbb{N}$ in $f(n)=\dfrac{1}{n}$ we obtain a real-sequence

n=1 f(1)=1

n=2 f(2)=1/2

n=3 f(3)=1/3

n=n f(n)=1/n

This real-sequence can be represented by

$\langle \dfrac{1}{n} \rangle := \langle 1, \dfrac{1}{2}, \dfrac{1}{3}, \ldots, \dfrac{1}{n}, \ldots$ .

1. $\langle {(-1)}^n \rangle$ is the sequence $\langle -1, 1, -1, 1, \ldots, {(-1)}^n, \ldots \rangle$ .
2. $\langle -3n \rangle$ is the sequence $\langle -3, -6, -9, \ldots, -3n, \ldots \rangle$
3. A sequence can also be formed by a recurrence relation with boundary values. If $f_n= f_{n-1}+f_{n-2} \ \text{for} n \ge 2$ and $f_0=f_1=1$ , then we obtain the sequence $\langle f_n \rangle$ as
n=1 $f_1=1$ (given)
n=2 $f_2=f_1 +f_0=1+1=2$ (given $f_0=1=f_1$ )
n=3 $f_3=f_2+f_1=2+1=3$
n=4 $f_4=f_3+f_2=3+2=5$
and so on…
This sequence, $\langle 1, 1,2, 3, 5, 8, 13, 21, \ldots \rangle$ is a real-sequence known as Fibonacci Sequence.

## Range Set of a Sequence

The set of all ‘distinct’ elements of a sequence is called the range set of the given sequence.

For example:

• The range set of $\langle \dfrac{1}{n}\rangle:= \{ \dfrac{1}{n} : n \in \mathbb{N} \}$ , which is an infinite set.
• The range set of $\langle {(-1)}^n \rangle := \{ -1, 1 \}$ , a finite set.

Remark: The range set of a sequence may be either infinite or finite, but a sequence has always an infinite number of elements.

## Sub-sequence of the Sequence

A sub-sequence of the sequence is another sequence containing some of the values of the sequence in the same order as in the original sequence. Alternatively, a sub-sequence of the sequence is another sequence which range set is a subset of the range set of the sequence.

For example:

• <1, 3, 5, 7, …> is a sub-sequence of the sequence <1, 2, 3, 4, …>.
• <1, 5, 13, 21, …> is a sub-sequence of the sequence <1,1,2,3,5,8,13,21, 34, …>.
• <1,1,1,1,1,…> is a sub-sequence of the sequence <-1, 1, -1, 1, …>. Since, the sequence <1,1,1,1,…> has only one value for each term, it’s called a constant sequence.

Remark: A sub-sequence is also a sequence hence it satisfy and follow all the properties of a sequence.

## Equality of two sequences

Two sequences $\langle S_n \rangle$ and $\langle T_n \rangle$ are said to be equal, if and only if $S_n=T_n, \forall n \in \mathbb{N}$ .

For example: The sequences $\langle \dfrac{n+1}{n} \rangle$ and $\langle 1+\dfrac{1}{n} \rangle$ are equal to each other.

Remark: From the definition the sequences <-1,1,-1,1, …> and <1,-1,1,-1,…> are not equal to each other, though they look alike and has same range set.

## Algebra of Sequences

Let $\langle S_n \rangle$ and $\langle T_n \rangle$ be two sequence, then the sequences having n-th terms $S_n+T_n, \ S_n-T_n, \ S_n \cdot T_n, \ \dfrac{S_n}{T_n}$ (respectively) are called the SUM, DIFFERENCE, PRODUCT, QUOTIENT of $\langle S_n \rangle$ and $\langle T_n \rangle$ .

For example: The sequence <1, 8, 19,30, …> is the sum of sequences <0, 1, 2, 3, …> and <1, 7, 17, 27, …> obtained after adding n-th term of one sequence to corresponding n-th term of other sequence. Similarly, other operations can be carried.

If $S_n \ne 0 \forall n$ , then the sequence $\langle \dfrac{1}{S_n} \rangle$ is known as the reciprocal of the sequence $\langle S_n \rangle$ .

For example: $\langle \dfrac{1}{1}, \dfrac{-1}{2}, \dfrac{1}{3}, \ldots \rangle$ is the reciprocal of the sequence $\langle 1, -2, 3, \ldots \rangle$ .

Remark: The sequences <-1,1,-1,1, …> and <1,-1,1,-1,…> have their reciprocals equal to the original sequence, hence these are called identity-sequences.

If $c \in \mathbb{R}$ then the sequence with n-th term $cS_n$ is called the scalar multiple of sequence $\langle S_n \rangle$ . This sequence is denoted by $\langle cS_n \rangle$ .

## Bounds of a Sequence

• A sequence $\langle S_n \rangle$ is said to be bounded above, if there exists a real number M such that $S_n \le M, \forall n \in \mathbb{N}$ . M is called an upper bound of the sequence $\langle S_n \rangle$ .
• A sequence $\langle S_n \rangle$ is said to be bounded below, if there exists a real number m such that $S_n \ge m, \forall n \in \mathbb{N}$ . m is called a lower bound of the sequence $\langle S_n \rangle$ .
• A sequence $\langle S_n \rangle$ is said to be bounded, if it is both bounded above and bounded below. Thus, if $\langle S_n \rangle$ is a bounded sequence, there exist two real numbers m & M such that $m \le S_n \le M \forall n \in \mathbb{N}$ .
• The least real number M, if exists, of the set of all upper bounds of $\langle S_n \rangle$ is called the least upper bound (supremum) of the sequence $\langle S_n \rangle$ .
• The greatest real number m, if exits, of the set of all lower bounds of $\langle S_n \rangle$ is called the greatest lower bound (infimum) of the sequence $\langle S_n \rangle$ .

Remark: If the range set of a sequence is finite, then the sequence is always bounded.

Examples:

• The sequence $\langle n^3 \rangle := \langle 1, 8, 27, \ldots \rangle$ is bounded below by 1, but is not bounded above.
• The sequence $\langle \dfrac{1}{n} \rangle := \langle 1, \dfrac{1}{2}, \dfrac{1}{3}, \ldots$ is bounded as it has the range set (0, 1], which is finite.
• The sequence $\langle {(-1)}^n \rangle := \langle -1, 1, -1, \ldots$ is also bounded.

# Convergent Sequence

A sequence $\langle S_n \rangle$ is said to converge to a real number l if for each $\epsilon$ >0, there exists a positive integer m depending on $\epsilon$ , such that $|S_n-l|$ < $\epsilon \ \forall n \ge m$ .

This number l is called the limit of the sequence $\langle S_n \rangle$ and we write this fact as $\lim_{n \to \infty} S_n=l$ and the sequence itself is called a convergent sequence. From now on, we’ll use $\lim S_n=l$ to represent $\lim_{n \to \infty} S_n=l$ , unless stated.

## Important Theorems on Convergent Sequences and Limit

1. (Uniqueness Theorem) Every convergent sequence has a unique limit.
2. For a sequence $\langle S_n \rangle$ of non-negative numbers, $\lim S_n \ge 0$ .
3. Every convergent sequence is bounded, but the converse is not necessarily true.
4. Let $\lim S_n= l$ and $T_n=l’$ , then $\lim (S_n +T_n) = l+l’$ , $\lim (S_n -T_n) = l-l’$ and $\lim S_n \cdot T_n = l \cdot l’$ .
5. Let $\langle S_n \rangle$ and $\langle T_n \rangle$ be two sequences such that $S_n \le T_n$ , then $\lim S_n \le \lim T_n$ .
6. If $\langle S_n \rangle$ converges to l, then $\langle |S_n| \rangle$ converges to |l|. In other words, if $\lim S_n = l$ then $\lim |S_n| =|l|$ .
7. (Sandwitch Theorem) If $\langle S_n \rangle$ , $\langle T_n \rangle$ and $\langle U_n \rangle$ be three sequences such that
1. $S_n \le T_n \le U_n, \ \forall n \in \mathbb{N}$
2. $\lim S_n=l= \lim U_n$ ,
then $\lim T_n=l$ .
8. (Cauchy’s first theorem on Limits) If $\lim S_n =l$ , then $\dfrac{1}{n} \{ S_1+S_2+ \ldots +S_n \} =l$ .
9. (Cauchy’s Second Theorem on Limits) If $\langle S_n \rangle$ is sequence such that $S_n$ > $0, \ \forall n$ and $\lim S_n =l$ , then $\lim {(S_1 \cdot S_2 \cdot \ldots S_n)}^{1/n}= l$ .
10. Suppose $\langle S_n \rangle$ is a sequence of positive real numbers such that $\lim \dfrac{S_{n+1}}{S-n} =l$ , ( l>0 ), then $\lim {(S_n)}^{1/n}=l$ .
11. (Cesaro’s theorem) If $\lim S_n=l$ and $\lim T_n=l’$ , then $\lim \dfrac{1}{n} \{ S_1 T_1 + S_2 T_2 + \ldots + S_n t_n \} = l \cdot l’$

# Theorem on Sub-Sequences

1. If a sequence $\langle S_n \rangle$ converges to l, then every subsequence of $\langle S_n \rangle$ converges to l, i.e., every sub-sequence of a given sequence converges to the same limit.

# Divergent Sequence

A sequence $\langle S_n \rangle$ is said to diverge if $\lim_{n \to \infty} S_n = +\infty$ or $\lim_{n \to \infty} S_n = -\infty$ .

# Oscillatory Sequence

• A sequence $\langle S_n \rangle$ is said to oscillate finitely if
I. It’s bounded.
II. It neither converges nor diverges.
• A sequence $\langle S_n \rangle$ is said to oscillate infinitely, if
I. It’s not bounded.
II. It neither converges nor diverges.

A sequence is said to be non-convergent if it’s either divergent or oscillatory.

# Limit Points of a Sequence

A real number P is said to be a limit point of a sequence if every neighborhood of P contains an infinite number of elements of the given sequence. In other words, a real number P is a limit point of a sequence $\langle S_n \rangle$ , if for a given $\epsilon$ >0, $S_n \in (P-\epsilon, P+\epsilon )$ for infinitely many values of n.

Bolzano Weierstrass Theorem: Every bounded real sequence has a limit point. (Proof)

Remarks:

• An unbounded sequence may or may not have a limit point.
• The greatest limit point of the bounded sequence $\langle S_n \rangle$ is called the limit superior of $\langle S_n \rangle$ and is denoted by $\lim \text{Sup} S_n$ .
• The smallest limit point of the bounded sequence $\langle S_n \rangle$ is called the limit inferior of $\langle S_n \rangle$ and is denoted by $\lim \text{Inf} S_n$ .
• limSup $\ge$ limInf.

# Monotonic Sequences

A sequence $\langle S_n \rangle$ is said to be monotonic if
either (i) $S_{n+1} \ge S_n, \forall n \in \mathbb{N}$
or, (ii) $S_{n+1} \le S_n, \forall n \in \mathbb{N}$ .

In first case, the sequence is said to be monotonically increasing while in the second case, it’s monotonically decreasing.

## Important Theorems on Monotonic Sequences

• A monotonically increasing sequence, which is bounded above, is convergent. (Otherwise, it diverges to $+\infty$ .) It converges to its supremum.
• A monotonically decreasing sequence, which is bounded below, is convergent. (Otherwise, it diverges to $-\infty$ .) It converges to its infimum.
• A monotonic sequence is convergent iff it’s bounded. (<== combination of first two theorems).

# Cauchy Sequences

A sequence $\langle S_n \rangle$ is said to be a Cauchy’s sequence if for every $\epsilon$ >0, there exists a positive integer m such that $|S_n -S_m|$ < $\epsilon$ , whenever $n \ge m$ .

## Important Properties of Cauchy Sequences

• Every Cauchy sequence is bounded. (proof)
• (Cauchy’s general principle of convergence) A sequence of real numbers converges if and only if it is a Cauchy sequence. (proof)

$\Box$