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

I Hyperreal Convergence

  1. Aug 11, 2017 #1
    I have always thought that non-constant sequences that converge toward 0 in the reals converge toward an infinitesimal in the hyperreals, but recently I have questioned my presumption. If ##(a_n)\to0## in ##R##, wouldn't the same seuqnece converge to 0 in ##*R##? These two statements should capture convergence of ##(a_n)\to 0## in the reals and hyperreals, respectively:

    (i) ##\forall \epsilon \in \mathbb{R}^+ \; \exists N \in \mathbb{N} \; \forall n \in \mathbb{N} : n \geq N \implies |a_n| < \epsilon##

    (ii) ##\forall \epsilon \in *\mathbb{R}^+ \; \exists N \in *\mathbb{N} \; \forall n \in *\mathbb{N} : n \geq N \implies |a_n| < \epsilon##

    For example, take ##(a_n=1/n)##. Clearly this converges to 0 in the reals. Choose a ##\epsilon \in *\mathbb{R}^+##; just for fun, say it is an infinitesimal. If ##\epsilon## is infinitesimal, then ##H=1/\epsilon## is hyperfinite. Any real number x has a natural number ##\lceil x \rceil## such that ##x \leq \lceil x \rceil < x+1##, by the transfer principle there much be a hypernatural ##\lceil H \rceil## such that ##H \leq \lceil H \rceil < H+1## and ##a_{\lceil H \rceil} \leq \epsilon##. Because the sequence is strictly decreasing, this means that all terms beyond ##\lceil H \rceil## will be strictly less than ##\epsilon##, and so the sequence must to 0 and not any infinitesimal. Is this correct? Thanks!
     
  2. jcsd
  3. Aug 12, 2017 #2

    Zafa Pi

    User Avatar
    Gold Member

    Though you and I have different notations/sources on non-standard analysis I'll give what I consider the best answer from my perspective since no one else is responding. If you merely say that the sequence {an} converges to 0 in R, then it converges to 0 whether you're thinking of the usual or non-standard reals.

    If you say, |ε| < 1/n for each standard n, then ε is not necessarily 0, it could be an infinitesimal. But that doesn't contradict the above.

    My understanding of non-standard analysis come from the 11 simple pages of Chapters 4, 5, and 6 of Ed Nelson's hyper-beautiful book: https://web.math.princeton.edu/~nelson/books/rept.pdf.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Hyperreal Convergence
  1. Uniform Convergence (Replies: 2)

  2. Sequence Convergence (Replies: 4)

  3. Convergent series (Replies: 3)

Loading...