A sequence is said to be Cauchy sequence if for given any integer n, there exists a positive real number R, such that for any n1, n2 > n, mod{n1th term - n2th term}
1,0,1,0,1,0,1,0........
Now for any integer n, whenever n1,n2 >n mod{n1th term - n2th term} < or equal to 1. So as per the definition of a Cauchy sequence we can say that this sequence is a Cauchy sequence, however, this is not a convergent sequence. How come? This implies there is gap in my understanding, can anyone kindly point out where am I wrong?.
Edited version:
A sequence $(a_n)$ is said to be Cauchy sequence if for given any integer $n$, there exists a positive real number $R$, such that for any $n_1, n_2 > n$, $|a_{n_1}-a_{n_2}|
We can prove that every Cauchy sequence is a convergent sequence.
Now let us consider the following sequence, $1,0,1,0,1,0,1,0,\ldots$
Now for any integer $n$, whenever $n_1,n_2 >n$ we have $|a_{n_1}-a_{n_2}|\le 1$ .
So as per the definition of a Cauchy sequence we can say that this sequence is a Cauchy sequence, however, this is not a convergent sequence. How come?
This implies there is gap in my understanding, can anyone kindly point out where am I wrong?.