What are the definitions of Cauchy sequence of real numbers and convergent sequence of real numbers?

If a sequence x_1, x_2, x_3... is convergent then it converges to some limit point x, meaning





For every e%26gt;0, there exists an N such that if n%26gt;N then |x_n - x| %26lt; e





Conceptually, this means that the terms of the sequence are getting closer and closer to their limit point.





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if a sequence x_1, x_2, x_3,... is cauchy, then





For every e%26gt;0 there exists an N such that if m%26gt;N and n%26gt;N then |x_n -x_m| %26lt; e





Conceptually, this means that the terms of the sequence are getting closer to each other. However, it's not just a term and it's successor that are getting close, it's a term and _every term after it_





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When you are dealing with the Real Numbers, it just so happens that a sequence is convergent if and only if it is cauchy. However this is not always the caseWhat are the definitions of Cauchy sequence of real numbers and convergent sequence of real numbers?
A sequence {x_n} is a Cauchy sequence if given any 蔚%26gt;0, there exists a positive integer N such that for m,n%26gt;N, |x_m-x_n|%26lt;蔚. Note that the limit of the sequence is not mentioned in this definition, and while if you have a Cauchy sequence of real numbers (using the usual distance), then the sequence does indeed have a limit, this is not necessarily true in sequences of other numbers.





A sequence converges to L if given 蔚%26gt;0, there exists a positive integer N, such that for all n%26gt;N, |x_n - L|%26lt;蔚.