The limit of a sequence theorems
      The cluster point or accumulation point
      Divergent sequences
      Sufficient condition for convergence of a sequence
         The Cauchy criterion (general principle of convergence)
The cluster point or accumulation point
Every sequence of numbers does not have to tend to a limit value. For example the terms of the sequence,
shown on the number line,
accumulate around two points, the odd terms around 0 while the even terms accumulate around 2. Therefore, it is the divergent sequence.
The cluster point is such a point of a sequence in every however small neighborhood of which lie infinitely many terms of the sequence. In the example above such points are 0 and 2.
According to this definition every limit point is the cluster point but inverse does not hold. 
That is, a convergent sequence can have at most one cluster point, hence if a sequence converges, then its limit is unique.
Divergent sequences
A sequence is convergent if it has a limit, otherwise it is divergent. A divergent sequence has no a finite limit.
The sequence of the natural numbers 1, 2, 3, . . . , n, . . .  has no a cluster point since it tends (or diverge) to infinity, so we write
Generally, if terms of a sequence {an} can become greater than every arbitrary large natural number N, that is, if for every given number N there exists an index n0 such that 
an > N   for all   n > n0(N),
we say that the sequence {an} tends to infinity or diverge to + oo  and we write either
an   + oo     as   n oo or    
If terms of a sequence {an} are such that for every given negative number - N, of arbitrary large absolute value, there exists an index n0 such that
an > - N   for all   n > n0(N),
we say that the sequence {an} tends to the negative infinity or diverge to  - oo  and we write either
an   - oo     as   n  oo or    
For example, the sequence {- n} tends to negative infinity or diverge to  - oo .
Sufficient condition for convergence of a sequence - The Cauchy criterion (general principle of convergence)
A sequence of real numbers, a1, a2, . . . , an, . . .  will have a finite limit value or will be convergent if for no matter how small a positive number e we take there exists a term an such that the distance between that term and every term further in the sequence is smaller than e, that is, by moving further in the sequence the difference between any two terms gets smaller and smaller.
As an + rwhere  r = 1, 2, 3, . . .  denotes any term that follows an,  then
| an + r - an | < e   for all  n > n0(e) r = 1, 2, 3, . . . 
shows the condition for the convergence of a sequence.
If a sequence {an} of real numbers (or points on the real line) the distances between which tend to zero as their indices tend to infinity, then {an} is a Cauchy sequence.
Therefore, if a sequence {an} is convergent, then {an} is a Cauchy sequence.
The Cauchy criterion or general principle of convergence, example
The following example shows us the nature of that condition.
Example:   We know that the sequence  0.3, 0.33, 0.333, . . .  converges to the number 1/3 as
1/3 = 0.33333 . . .  .   Let write the rule for the nth term,
If we go along the sequence far enough, say to the 100th term, i.e., the term with a hundred 3's in the fractional part, then the difference between that term and every next term is equal to the decimal fraction with the fractional part that consists of a hundred 0's followed by 3's on the lower decimal places, starting from the 101st decimal place. That is,
Therefore, the absolute value of the difference falls under
Then, if we go further along the sequence and for example calculate the distance between the 100000th term
and the following terms, the distance will be smaller than
Hence, since we can make the left side of the inequality  | an + r - an | < e  as small as we wish by choosing n large enough, then all terms that follow an (denoted  an + r,   r = 1, 2, 3, . . .  ), infinitely many of them, lie in the interval of the length 2e symmetrically around the point an. Outside of that interval there is only a finite number of terms. That is,
 - e < an + r - an < + e   for all  n > n0(e) r = 1, 2, 3, . . . 
                      or            an - e < an + r < an + e. 
So, the terms of the sequence, starting from the (n + 1)th term, form the infinite and bounded sequence of numbers and so, according to the above theorem, they must have at least one cluster point that lies in that interval. But they cannot have more than one cluster point since all points that follow the nth term lie inside the interval 2e length of which is arbitrary small, if n is already large enough, so that any other cluster point will have to be outside of that interval.
Thus, the above theorem simply says that if a sequence converges, then the terms of the sequence are getting closer and closer to each other as shows the example.
Calculus contents B
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