## Sequence of real numbers

A sequence of real numbers (or a real sequence) is defined as a function , where is the set of natural numbers and is the set of real numbers. Thus, is a function which produces a sequence of real numbers . It’s customary to write a sequence as form of functions in brackets, e.g.; , . We can alternatively represent a sequence as the function with natural numbers as subscripts, e.g., , . This alternate method is a better representation of a sequence as it distinguishes ‘a sequence’ from ‘a function’. We shall use notation and when writen , we mean a sequence with infinitely many terms. Since all of are real numbers, this kind of sequence is called a sequence of real numbers.

## Examples of Sequences

- Like is a real-valued-function, is a real sequence.

Putting consecutive values of

in 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

.- is the sequence .
- is the sequence
- A sequence can also be formed by a recurrence relation with boundary values. If and , then we obtain the sequence as

*n=1*(given)

*n=2*(given )

*n=3*

*n=4*

and so on…

This sequence, 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 , which is an infinite set.
- The range set of , 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

and are said to be equal, if and only if .For example: The sequences

and 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

and be two sequence, then the sequences having*n*-th terms (respectively) are called the SUM, DIFFERENCE, PRODUCT, QUOTIENT of and .

*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

, then the sequence is known as the*reciprocal*of the sequence .

*For example: *

* 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

then the sequence with*n*-th term is called the scalar multiple of sequence . This sequence is denoted by .

## Bounds of a Sequence

- A sequence is said to be
, if there exists a real number**bounded above***M*such that .*M*is called an upper bound of the sequence . - A sequence is said to be
, if there exists a real number**bounded below***m*such that .*m*is called a lower bound of the sequence . - A sequence is said to be
**bounded***,*if it is both bounded above and bounded below. Thus, if is a bounded sequence, there exist two real numbers*m & M*such that . - The least real number
*M,*if exists, of the set of all upper bounds of is called the least upper bound (supremum) of the sequence . - The greatest real number
*m*, if exits, of the set of all lower bounds of is called the greatest lower bound (infimum) of the sequence .

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

Examples:

- The sequence is bounded below by 1, but is not bounded above.
- The sequence is bounded as it has the range set (0, 1], which is finite.
- The sequence is also bounded.

## Convergent Sequence

A sequence

is said to converge to a real number*l*if for each >0, there exists a positive integer

*m*depending on , such that < .

This number *l* is called the limit of the sequence

## Important Theorems on Convergent Sequences and Limit

- (
*Uniqueness Theorem*) Every convergent sequence has a unique limit. - For a sequence of non-negative numbers, .
- Every convergent sequence is bounded, but the converse is not necessarily true.
- Let and , then , and .
- Let and be two sequences such that , then .
- If converges to
*l*, then converges to |*l*|. In other words, if then . - (
*Sandwitch Theorem*) If , and be three sequences such that- ,

then .

- (
*Cauchy’s first theorem on Limits*) If , then . - (
*Cauchy’s Second Theorem on Limits*) If is sequence such that > and , then . - Suppose is a sequence of positive real numbers such that , (
*l>0*), then . - (
*Cesaro’s theorem*) If and , then

## Theorem on Sub-Sequences

- If a sequence converges to
*l*, then every subsequence of converges to*l*, i.e., every sub-sequence of a given sequence converges to the same limit.

## Divergent Sequence

A sequence

is said to diverge if or .## Oscillatory Sequence

- A sequence is said to oscillate finitely if

I. It’s bounded.

II. It neither converges nor diverges. - A sequence 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

*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 is called the
*limit superior*of and is denoted by . - The smallest limit point of the bounded sequence is called the
*limit inferior*of and is denoted by . - limSup limInf.

## Monotonic Sequences

A sequence

is said to be monotonic ifeither (i)

or, (ii) .

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

## Cauchy Sequences

A sequence

is said to be a Cauchy’s sequence if for every >0, there exists a positive integer*m*such that < , whenever .

## 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)