Module 1 : Real Numbers, Functions and Sequences
Lecture 3 : Monotone Sequence and Limit theorem [ Section 3.1 : Monotone Sequences ]
 
3.1.3 Theorem:
  If  is monotonically decreasing and is bounded below, it is convergent.
  
Proof :
 

Follows from the following facts.

(i)  is monotonically decreasing if and only if  monotonically increasing.
 (ii)  is bounded below if and only if is bounded above.
  (iii) is convergent if and only if is convergent .
   
   
 

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