
Limits of sequences 
Properties of
convergent sequences 
The limit value is exclusively determined by the behavior of the terms
in its close neighborhood 
Bounded
sequences 






Properties of
convergent sequences 
The limit value is exclusively determined by the behavior of the terms
in its close neighborhood 
a)
We
use
the definition of convergence that states, a
number L is the limit of a sequence
if
for every e
> 0
there
exist a natural number n_{0}(e)
such that 
 a_{n}

L  < e
for all n >
n_{0}(e), 
to
examine the behavior of the sequence {a_{n}}
as n
tends
to infinity. 
Considering
the geometric interpretation of the
above inequality we can write 

e
< a_{n}

L
< +
e
for all n >
n_{0}(e), 
or
L

e
< a_{n}
< L
+ e
for all 
shown
on the number line (see the
example sequence (5)
in the previous section) 

Thus, if
a number L
is the limit of a sequence {a_{n}}
then in the interval from L

e
to L
+ e,
of the length 2e,
symmetrical with respect to L,
lie all terms of the sequence beginning with the one which is far
enough, indexed by n_{0},
so lie infinitely many terms, and it holds to every however small
interval 2e.
This (infinite) part
of the length 2e
of a convergent sequence is usually called tail. 
Outside
of this interval there is finite but any number of initial terms of the
sequence, but they don't have any influence
on existence of the limit or its value. The value of the limit stays
unchanged if these initial terms are,
substituted, dropped or are replaced by another. Thus, the beginning
part of the convergent sequence outside
of the interval 2e
is a finite or discrete set of numbers. 
Therefore,
the limit value is exclusively determined by the behavior of the terms
in its close neighborhood. 

Bounded
sequences 
A
sequence is bounded above if there is a number M
such that a_{n}
< M for all n. 
It is bounded below if there is
a number m
such that a_{n}
> m for
all n. 
If
a sequence is bounded above and bellow it is called bounded sequence. 
Thus,
a sequence is bounded if there exist a number M > 0
such that  a_{n}  <
M
for
all n. 
A
sequence which is not bounded is called unbounded. 

b)
From the fact that outside the interval
of the length 2e
lie only finite number of terms of a sequence it follows
that every convergent sequence is bounded, that is 
there exists
a
number M > 0
such that  a_{n}  <
M
for
all n. 
Therefore, all terms of a convergent sequence can be closed, in the interval
containing finite number of the initial
terms outside the interval 2e,
and in the interval 2e
itself. 
Any
convergent sequence is bounded (both above and below). 

c)
If
{a_{n}}
is a convergent sequence, then every subsequence of that sequence
converges to the same limit. 
Let
from a convergent sequence extracted is infinitely many terms, a_{n}_{1},
a_{n}_{2},
. . . ,
a_{n}_{p},
. . . ,
using any principle, for example
extracted is every other term, then
they make new infinite sequence called subsequence.
Then, the extracted
subsequence
converges to the same limit as the original sequence. 
That
is, if  a_{n}

L  < e
for all n >
n_{0}(e),
then  a_{n}_{p}

L  < e
for all n_{p}
>
n_{0}(e) 
since the natural numbers n_{p}
are contained within n,
and n_{p}
®
oo when
n
®
oo
. 

d)
The
limit L
can belong to a sequence but should not. By its definition the limit
does not belong to a
sequence since the definition says, the limit is a value that is
approached increasingly closely by a sequence as n
tends to infinity. 

e)
Terms
of a sequence which are far
enough can be considered as the approximate value of the limit L. 
This
way defined is every irrational number as the limit of the two
infinite sequences, {a_{n}}
and {b_{n}}
of rational
numbers that satisfies following conditions. 
1)
The sequence {a_{n}}
is increasing (nondecreasing) and the {b_{n}}
is decreasing (nonincreasing), i.e., 
a_{1}
<
a_{2}
<
. . . <
a_{n}
<
a_{n }_{+
1}
<
. . .
and b_{1}
>
b_{2}
>
. . . >
b_{n}
>
b_{n }_{+
1}
>
. . . 
2)
None of the terms of {a_{n}}
is greater then any of the terms of {b_{n}}, 
a_{n}
<
b_{n}
where n
= 1, 2, 3, .
. . 
that
is, every number a_{n}
lies to the left of every number b_{n}
on the number line. 
3)
The difference b_{n}

a_{n}
of a two terms of the given sequences of the same index n
can become arbitrary
small as n
tends to infinity, 
lim
(b_{n}

a_{n})
= 0 as
n
®
oo 
as
is shown on the number line below. 

Therefore,
according
to the given conditions the following must hold. 
There
is only one real number L
determined by these two sequences such that none of the terms of {a_{n}}
is
greater then L
(a_{n}
<
L)
and that none of the terms of {b_{n}}
is less then L
(b_{n}
>
L). 
Thus, both sequences,
{a_{n}}
and {b_{n}}
approach the
same limit L
from below and above, i.e., 

For
example, the circumference of the circle is the limit to which
the circumferences of inscribed and circumscribed
polygons increase and decrease, respectively, as the number of sides
increases to infinity. 











Calculus
contents B 



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