Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Extended Real definition of Cauchy sequence?

  1. Sep 16, 2010 #1
    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?
  2. jcsd
  3. Sep 17, 2010 #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.
  4. Sep 17, 2010 #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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Extended Real definition Date
Extended divergence theorem Sep 18, 2013
Extending Taylor expansions Oct 29, 2012
Extending radius of convergence by analytic continuation Mar 1, 2012
Infimum and extended real number May 3, 2011
Extended real-valued function Jun 28, 2010