# Real Sequences

Posted on the 12 August 2013 by Gaurav Tiwari @wpgaurav

## 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

1. 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

.

1. is the sequence
.
2. is the sequence
3. 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:

is the reciprocal of the sequence
.

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 bounded above, if there exists a real number M such that
. M is called an upper bound of the sequence
.
• A sequence
is said to be bounded below, if there exists a real number 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

and we write this fact as
and the sequence itself is called a convergent sequence. From now on, we’ll use
to represent
, unless stated.

## Important Theorems on Convergent Sequences and Limit

1. (Uniqueness Theorem) Every convergent sequence has a unique limit.
2. For a sequence
of non-negative numbers,
.
3. Every convergent sequence is bounded, but the converse is not necessarily true.
4. Let
and
, then
,
and
.
5. Let
and
be two sequences such that
, then
.
6. If
converges to l, then
converges to |l|. In other words, if
then
.
7. (Sandwitch Theorem) If
,
and
be three sequences such that
1. ,
then
.
8. (Cauchy’s first theorem on Limits) If
, then
.
9. (Cauchy’s Second Theorem on Limits) If
is sequence such that
>
and
, then
.
10. Suppose
is a sequence of positive real numbers such that
, ( l>0 ), then
.
11. (Cesaro’s theorem) If
and
, then

## Theorem on Sub-Sequences

1. 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’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

, if for a given
>0,
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
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 if
either (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)