Extended Real definition of Cauchy sequence?

In summary, the conversation discusses the concept of a Cauchy sequence and the requirement for an extended definition, including a proof that a sequence diverging to infinity is Cauchy and vice versa. The speakers explore various definitions and potential solutions, including one involving a sequence of complex numbers and the need to consider both positive and negative infinity.
  • #1
joeblow
71
0
Is there an extended definition of a Cauchy sequence? My prof wants one with a proof that a sequence divergent to infinity is Cauchy and vice versa.

My first thought was that a sequence should be Cauchy if it is Cauchy in the real sense or else that for any M, there are nth and mth terms of the sequence such that that their difference is g.t. M and the difference between the nth and (m+j)th terms is g.t. M for j = 0,1,... . But this definition is dependent on whether the sequence approaches + or - infinity.

Any ideas?
 
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  • #2
Try this : for each M>0 there exists m such that |An+j - An|> M for all n >= m. This will make a sequence An of complex numbers 'converge' to infinity.
 
  • #3
That's very close to what I have, except that we are dealing with reals, so that definition would treat a sequence that has a subsequence converging to + infinity and a subsequence convergent to - infinity as Cauchy, which I don't think is acceptable. It is in C since there's just one infinity.
 

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of real numbers in which the elements of the sequence become arbitrarily close to one another as the sequence progresses. In other words, for any given small distance, there exists a point in the sequence where all subsequent elements are within that distance from each other.

2. What is the difference between Cauchy sequence and convergent sequence?

A Cauchy sequence is a type of sequence in which the elements become arbitrarily close to each other, whereas a convergent sequence is a type of sequence in which the elements approach a specific limit as the sequence progresses. Not all Cauchy sequences are convergent, but all convergent sequences are considered Cauchy sequences.

3. How is the Extended Real definition of Cauchy sequence different from the traditional definition?

The traditional definition of a Cauchy sequence only applies to sequences of real numbers, while the Extended Real definition allows for sequences of extended real numbers, which include positive and negative infinity. This allows for a more general definition that can be applied to a wider range of sequences.

4. What is the importance of Cauchy sequences in mathematics?

Cauchy sequences are important in mathematics because they are used to define the notion of a complete metric space, which is a fundamental concept in analysis. They also play a key role in the construction of real numbers from rational numbers, and are used in many proofs and theorems in various branches of mathematics.

5. Can a Cauchy sequence have more than one limit?

No, a Cauchy sequence can only have one limit, which is the real number that the elements of the sequence approach as the sequence progresses. If a Cauchy sequence has more than one limit, then it is not a valid Cauchy sequence and does not satisfy the definition.

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