Module 1 : Real Numbers, Functions and Sequences
Lecture 2 : Convergent & Bounded Sequences [ Section 2.1 : Need to consider sequences ]
 
 

To understand this paradox, let us assume that the man walks with a constant speed. Suppose he takes minutes to cover the first half of the distance. The next half will be covered in minutes, the half of the remaining half in minutes, and so on. The time consumed at the th stage will be
                                    .
This gives us the sequence ,  , . . . . The paradox is that there are infinite stages and how they can covered in finite time? The paradox is resolved by proper interpretation of 'infinite'.

Let us observe that in both the examples, we are interested in finding out what happens to the sequence of observations for large ? To analyze this problem, let us make some definitions.
2.1.3   Definition :
  A sequence of elements of a set is an ordered collection : ,    , . . . .  ,  . . . of elements of . The element  is called its first term,  - its second term , and in general  as its term. We also write this as . One can also think of a sequence  in as a function f defined on the set of natural numbers with values in , i.e., : with . Note that a sequence is not the same as the set {, , . .  } .
 
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