Give an example of a sequence {an} such that it passes three limit points.[Hint:Example of a sequence {an} passes two limits is {(-1)^n}]

1,-1,1,1,-1,1/2, 1,-1,1/3,....

 

i.e. the sequence given by

 

a_n = 1 if n=3k+1

    = -1 if n = 3k+2

    = 1/k if n = 3k

has infinitely many terms that are in any neighbourhood of 1,-1, and 0. Hence this sequence has 3 limit points

can you please explain it......

First off, a limit point for a real number sequence is a real number that has infinitely many terms at any distance from it (i.e. in any epsilon neighbourhood). This is a relaxed condition from the concept of a limit, which requires that all terms excepting a few finite number of them should be found in any epsilon neighbourhood.

 

So, in the sequence above, if you take 0, in any epsilon neighbourhood, you will find infinitely many terms of the type 1/k at any distance from 0. Hence 0 is a limit point. The arguments for 1 and -1 are similar.

 

By the way, this is not needed for JEE. It comes from a topic called ‘Topology’ applied to real numbers.