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
         Every convergent sequence is bounded
         Increasing, decreasing, monotonic sequence
         Every subsequence of a convergent sequence converges to the same limit
         Every bounded monotonic sequence is convergent example
Properties of convergent sequences
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.
Increasing, decreasing, monotonic sequence
A sequence is increasing (or strictly increasing) if
a1 < a2 < . . . < an < an + 1 < . . . ,   or   an + 1 > an  for every n.
A sequence is decreasing (or strictly decreasing) if
a1 > a2 > . . . > an > an + 1 > . . . ,   or   an + 1 < an  for every n.
A sequence is called monotonic (monotone) if it is either increasing or decreasing.
Thus, increasing sequences either, diverge to + oo  as n tends to infinity, written
(as, for example, the sequence of natural numbers) or it is bounded above,  an < for all n N, that is all terms of the sequence remain less than a fixed number M.
Decreasing sequences either, diverge to  - oo  as n tends to infinity, written
(as, for example, the sequence -1, -2, -3, . . .   ) or it is bounded below, an > for all n N, that is all terms of the sequence remain greater than a fixed number m.
Monotonic sequence theorem
An increasing sequence, whose all terms are less than a fixed number M, tends to a finite limit value L that is not greater than M, therefore
if   an + 1 > an    and    an < M,   n = 1, 2, 3, . . .  then    
A decreasing sequence, whose all terms are greater than a fixed number m, tends to a finite limit value L that is not less than m, therefore
if   an + 1 < an     and    an > m,   n = 1, 2, 3, . . .  then    
Thus, every bounded monotonic sequence is convergent.
Example:  Prove that the sequence given by     is increasing and bounded.  
Solution:  Since  
that is,  an + 1 > an  for any natural number n, therefore the sequence is increasing.
By drawing the first few terms of the sequence on the number line and examining the nth term expression as n tends to infinity, it follows that given sequence is bounded above by 1,
Let verify that 1 is the limit of the sequence using the definition,
therefore, as we can make the distance between the nth term and 1 as small as we wish by choosing n large enough, given sequence is convergent.
Calculus contents B
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