Module 1 : Real Numbers, Functions and Sequences
Lecture 3 : Monotone Sequence and Limit theorem [ Section 3.3 : Some Extensions of the Limit Concept ]
3.3 Some Extensions of the Limit concept:
3.3.1 Definition:
  Let   be a sequence of real numbers.                  
(i) 


 We say    converges to   if for every    ,    is ultimately bigger than   ,  i.e., given    such that   . We write this as .
(ii)


 We say   converges to    if for every ,   is ultimately smaller than , i.e., given    such that  .   We write this as  .
(iii) We say a sequence is divergent properly if either it converges to   or it converges to  .
3.3.2
 Examples:
(i) Consider the sequence . Given any , by the Archimedian property, we can find positive integer N

 

 such that .  Thus, for every . Hence, .
(ii)
 Let , .Consider the sequence . We shall show that . To see this let
      , where .  Using Binomial theorem,
                                 
Once again, using the Archimedian property, we can find positive integer N  such that . Then, by the above inequality, we get  for every . Hence, .  Here are some intuitively obvious results.
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