state and prove cauchy's general principle of convergence .
Answers
A sequence of real numbers, a1, a2, . . . , an, . . . will have a finite limit value or will be convergent if for no matter how small a positive number e we take there exists a term an such that the distance between that term and every term further in the sequence is smaller than e, that is, by moving further in the sequence the difference between any two terms gets smaller and smaller.
As an + r, where r = 1, 2, 3, . . . denotes any term that follows an, then
| an + r - an | < e for all n > n0(e), r = 1, 2, 3, . . .
shows the condition for the convergence of a sequence.
If a sequence {an} of real numbers (or points on the real line) the distances between which tend to zero as their indices tend to infinity, then {an} is a Cauchy sequence.
Therefore, if a sequence {an} is convergent, then {an} is a Cauchy sequence