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

Question Re. Simple epsilon proofs

  1. Aug 30, 2009 #1
    Here's an example that helps illustrate my question:

    Prove: A sequence in R can have at most one limit.


    Assume a sequence {xn}n[tex]\in[/tex]N has two limits a and b.
    By definition:

    -For any [tex]\epsilon[/tex]>0, there exists an N[tex]\in[/tex]N such that n[tex]\geq[/tex]N implies that |xn-a| < [tex]\epsilon[/tex]/2.
    -A similar argument can be made for the limit b.


    |a-b| [tex]\leq[/tex] |xn-a| - |xn-b| < [tex]\epsilon[/tex]/2 + [tex]\epsilon[/tex]/2 = [tex]\epsilon[/tex]

    Thus a=b.

    Now here's my question...in the step immediately following "By definition," the actual definition of a limit of a sequence shows that n[tex]\geq[/tex]N implies that |xn-a| < [tex]\epsilon[/tex], not that n[tex]\geq[/tex]N implies that |xn-a| < [tex]\epsilon[/tex]/2. So is it acceptable to put [tex]\epsilon[/tex] over any number when convenient in a proof? As in the above proof, it is convenient to put [tex]\epsilon[/tex]/2 instead of just [tex]\epsilon[/tex] so that in the final step the two add up to [tex]\epsilon[/tex] and show that a=b.


    edit: sorry for the awkward formatting, if anything is unclear, let me know I'll explain it in words.
    Last edited: Aug 30, 2009
  2. jcsd
  3. Aug 30, 2009 #2
    Yes, if the limit exists, then you can replace epsilon with any positive number (so epsilon/2 is of course positive), because that's simply what epsilon typically denotes. Many people want the last step of a proof to be just epsilon, but often times it doesn't matter. In this case, it doesn't matter whether we have epsilon or 2*epsilon at the very end, since epsilon is arbitrary anyways. Note you have a typo in the leftmost inequality at the final step of the proof.
  4. Aug 30, 2009 #3
    Thanks for the reply, that's what I thought but wanted to hear it from someone who actually knew before I went on doing proofs on an assumption.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook