# 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

- 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$ .

- $ \langle {(-1)}^n \rangle$ is the sequence $ \langle -1, 1, -1, 1, \ldots, {(-1)}^n, \ldots \rangle$ .
- $ \langle -3n \rangle$ is the sequence $ \langle -3, -6, -9, \ldots, -3n, \ldots \rangle$
- 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
, if there exists a real number**bounded above***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
, if there exists a real number**bounded below***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

- (
*Uniqueness Theorem*) Every convergent sequence has a unique limit. - For a sequence $ \langle S_n \rangle$ of non-negative numbers, $ \lim S_n \ge 0$ .
- Every convergent sequence is bounded, but the converse is not necessarily true.
- 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’$ .
- 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$ .
- 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|$ . - (
*Sandwitch Theorem*) If $ \langle S_n \rangle$ , $ \langle T_n \rangle$ and $ \langle U_n \rangle$ be three sequences such that- $ S_n \le T_n \le U_n, \ \forall n \in \mathbb{N}$
- $ \lim S_n=l= \lim U_n$ ,

then $ \lim T_n=l$ .

- (
*Cauchy’s first theorem on Limits*) If $ \lim S_n =l$ , then $ \dfrac{1}{n} \{ S_1+S_2+ \ldots +S_n \} =l$ . - (
*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$ . - 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$ . - (
*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

- 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’sbounded.**not**

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$

You can either start a new conversation or continue an existing one.Please don't use this comment form just to build backlinks. If your comment is not good enough and if in some ways you are trying to just build links — your comment will be deleted. Use this form to build a better and cleaner commenting ecosystem. Students are welcome to ask for help, freebies and more. Your email will not be published or used for any purposes.