Extended Real definition of Cauchy sequence?

[joeblow]

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?

 

[Eynstone]

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.

 

[joeblow]

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.