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 n0(e) such that
| an - L | < e  for all  n > n0(e),
to examine the behavior of the sequence {an} as n tends to infinity.
Considering the geometric interpretation of the above inequality we can write
 - e < an - L < + e   for all  n > n0(e),
                                 or                L - e < an < 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 {an} 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 n0, 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 an < M  for all n.
It is bounded below if there is a number m such that  an > 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  | an | < 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  | an | < 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 {an} 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, an1, an2, . . . , anp, . . . , 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  | an - L | < e  for all  n > n0(e),  then  | anp - L | < e  for all  np > n0(e)
since the natural numbers np are contained within n, and  np 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, {an} and {bn} of rational numbers that satisfies following conditions.
1)   The sequence {an} is increasing (nondecreasing) and the {bn} is decreasing (nonincreasing), i.e., 
a1 < a2 <  . . . < an < an + 1 < . . .    and    b1 > b2 > . . . > bn > bn + 1 > . . . 
2)   None of the terms of {an} is greater then any of the terms of {bn},   
 an < bn  where  n = 1, 2, 3, . . .
that is, every number an lies to the left of every number bn on the number line. 
3)   The difference bn -  an of a two terms of the given sequences of the same index n can become arbitrary small as n tends to infinity,
lim (bn -  an) = 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 {an} is greater then L (an < L) and that none of the terms of {bn} is less then L (bn > L).
Thus, both sequences, {an} and {bn} 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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