Introduction to Real Sequences
George Cantor (1845—1918), the creator of the set theory, made considerable contributions to the
development of the theory of real sequences. He found a firm base for most of the fundamental concepts of
real analysis in the sequences of rational numbers. Though his layouts are not convenient in the initial
stages, they are quite advantageous while making advanced investigations. The study of many important
and advanced concepts becomes easy if the notion of sequences is employed.
A sequence is a function whose domain set is the set , whereas the range set may be any set. Now
onwards, we shall deal with those specific sequences whose range sets sub-sets of . Such sequences are
called real sequences. Thus, the function
is a real sequence.
Sequences:
A function whose domain is the set of natural numbers and ranges a subset of is a real
sequence or simply a sequence. Symbolically, if
then is a sequence. As in the case of
functions, we denote a sequence in a number of ways. Usually a sequence is denoted by its images. For a
sequence , the image corresponding to
is denoted by
or
is called the nth term (or
member or element) of the sequence .
The set of all distinct terms of a sequence is called therange set of that sequence, we shall denote
the range set of a sequence by
or by
.
Since the domain set for a sequence is always , if we could characterize the nth
term of a sequence then it evidently fully defines the sequence. Thus we shall denote a
sequence, , by any one of
, or
simply by
where
the nth term is supposed to be known. The nth term , is either directly known or it is given by specifying
some relations from which it could be determined for each
given then the recurrence relation
. For example, if
and
are
specifies (since in succession
could be determined). Thus the sequence
is fully
prescribed.
Note that for writing the general terms of a given sequence, one can start from any stage from where
u appears to be generated correspondingly. For example, for the sequence
one can take the general term as
respectively.
. Here for
we get 3rd, 4th, 5th... terms
Constant Sequences:
A sequence defined by
is called a constant sequence. When there is no
ambiguity the number is itself used to signify this constant sequence .
Boundedness of Sequences
A sequence
that
.
or
is said to be bounded below or above accordingly as there exist numbers
,
. The numbers
and
such
are known as the lower andupper bounds of
A sequence bounded below as well as above is said to be a bounded sequence; otherwise, if it is
either unbounded below or above, or both then it is said to be anunbounded sequence. Thus is bounded
if
and
such that
that
,
. Equivalently,
is bounded if
such
.
Evidently,
is bounded if and only if
is bounded. Upper and lower bounds (supremum and
infimum) of
, if exist, are called the upper and lower bounds of the sequence . A constant sequence
is obviously bounded.
In sequences, terms with equal values can occur. Therefore, a sequence may have more than one
term with the smallest value. In such a case any of these is taken for the smallest value. In fact while
talking about the smallest term we are substantially interested in the value of the term rather than the
position of the term in the sequence. Similar explanation holds for the greatest term. Note that, like sets of
real numbers, a sequence bounded below or above may or may not have a smallest or a greatest member
accordingly. Clearly, an unbounded sequence cannot have a smallest or a greatest member.
Example: The sequence whose nth term is.
(i)
is bounded and has smallest and greatest terms 0, 2. Every non-positive number is a lower
bound and any member of
is an upper bound of the sequence.
(ii)
is bounded below and has smallest term as 2. Every member of
sequence and the sequence is unbounded above.
(iii)
(iv)
and
, is a lower bound of the
n is unbounded, both ways.
is bounded and has no smallest member whereas it has 1 as the greatest member
are its sets of lower and upper bounds.
Limit Points of a Sequence
A number
that
is said to be a limit point of a sequence
, for infinitely many values of
if every neighborhood
, i.e. for any
, of
,
is such
, for finitely many
values of
. Evidently, if
, for infinitely many values of then is a limit point of the
sequence .
As in the case of sets of real numbers, limit points of a sequence may also be called accumulation,
cluster or condensation points. The limit points of a sequence may be classified in two types: (i) those for
which
, for infinitely many values of
, (ii) those for which
for only a finite number of
values of
. But this distinction not very much needed. As such, we do not distinguish the above
mentioned two types of limit points of sequences by different titles. It should be noted that every limit
point of the range set
neighborhood of
of a sequence
is also a limit point of the sequence
contains infinitely many points of
, because every
and so of the sequence . On the other hand, a
limit point of may or may nor be a limit point of
.
If the values of only a finite number of terms of are not distinct then, evidently the limit points
of
are the same as those of the set
. Conclusively, it follows that the limit points of a sequence
are either the points or the limit points of the set
Example 1: If a sequence
sequence.
Solution: For any
is defined by
,
.
, then
is the only limit point of
. Therefore,
is a limit pint of the sequence.
Let
and
. Then for all ,
. When
. Thus no point
other than is the limit point of the sequence. Note that the limit point of the sequence is not a limit
point of the range
Example 2: If
.
, then
is the only limit point of the sequence .
Sufficient conditions for number to be or not to be a limit point of a sequence .
1. If for every
such that
,
or equivalently
, then is a limit point of the sequence .In such a case it can be easily seen that is the
only limit point of the sequence. The above condition is not necessary as it can be seen for the
sequence
satisfied.
2. If for an
,
,
is a limit point of this sequence but the above condition is not
for only a finite number of values of
then is not a limit
point of . Such a condition is also necessary for a number not to be a limit point of the
sequence .
Remarks:
1. Whenever we simply write
it is implied that may be howsoever small positive number.
2. A positive number is said to be arbitrary small if given any
, may be chosen such
that
3. If
.
be an arbitrary small positive number and given any
then
positive number. This follows immediately if we take
4. If
sequence
is also an arbitrary small
for an
.
are two arbitrary small positive numbers then it readily follows that is a limit point of a
if and only if
for infinitely many values of .
Example 3: Every bounded sequence
has at least one limit point.
Example 4: The set of limit points of a bounded sequence
Theorem: The set of limit points
Corollary: Every bounded set
of every sequence
is bounded.
is a closed set.
, of limit points of a sequence , contains smallest and greatest members.
Upper and Lower Limits of a Bounded Sequence
The greatest and smallest limit points of a bounded sequence, as given by the preceding tutorial are
respectively called the upper (or superior) and lower (or inferior) limits of the sequence.
The upper limit of a bounded sequence
limit is denoted by
or
For bounded
is denoted by
. Evidently,
, the limits
or
. Similarly, the lower
.
shall also be defined as
,
Where
are defined by
,
.