Module 1 : Real Numbers, Functions and Sequences
Lecture 3 : Monotone Sequence and Limit theorem [ Section 3.1 : Monotone Sequences ]
 
3.1
  Monotone Sequences
3.1.1
  Definition:
(i) A sequence said to be monotonically increasing, if   
(ii) A sequence  said to be monotonically decreasing, if   
  We can describe now the completeness property of the real numbers.
3.1.2
 Completeness property
Every monotonically increasing sequence which is bounded above is convergent.
3.1.3
 Theorem:
If  is monotonically decreasing and is bounded below, it is convergent.                       
3.1.4
 Examples:
   (i)   Sequence is monotonically increasing and is not bounded above.
(ii)  Sequence is monotonically decreasing and is bounded below,say by .
(iii)  Sequence is neither monotonically increasing nor decreasing.
(iv)

Let      As shown in problem 2.1(Lecture 1) ,   and
. Hence,  is convergent. Let    .  Then,
                                         
Hence,   . This implies that  . Hence,

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