Module 1 : Real Numbers, Functions and Sequences
Lecture 3 : Monotone Sequence and Limit theorem [ Section 3.2 : Limit Theorems on Sequences ]
 
   
  Additional Remarks : 
   (i)






 In Practice exercise 2(i) of section 1.6, we defined   
The sequence  is a monotonically decreasing sequence of rational numbers which is bounded below. However, it cannot converge to a rational (why?). This exhibits the need to enlarge the concept of numbers beyond rational numbers. The sequence  converges to and its elements are used to find rational approximation (in computing machines) of .
(ii)


 		 To prove that a sequence is convergent to , one needs to find a real number (not given by the sequences) and  verify the required property. However, the concept of 'Cauchyness' of a sequence is purely an 'intrinsic' property which can be verified purely by the given sequence. Still a sequence is Cauchy if and only if it is convergent.
 (iii) 


 Using the completeness property we can say that the sequence of the areas of -sided regular polygons inside the unit circle is an increasing sequence which is bounded above. Its limit is denoted by ,
called pi. This gives a definition of . It is also an irrational number.
 
   Optional Exercises : 
 
(1) 

 Let  be a sequence and let
(i) 
 Show that is convergent to , whenever is convergent to .
(ii) 
 Given an example to show that the converse of (i) need not be true.
(2) 
 Prove that the sequence is convergent as follows:
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