1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove alpha=sup(S) is equivalent to alpha belongs to S closure

  1. Oct 12, 2010 #1
    Given that alpha is an upper bound of a given set S of real numbers, prove that the following two conditions are equivalent:
    a) We have alpha=sup(S)
    b) We have alpha belongs to S closure

    I'm trying to prove this using two steps.
    Step one being: assume a is true, then prove b is true.
    Step two being: assume b is true, then prove a is true.

    Could anyone help me with step two?
    Assuming alpha belongs to S closure.....
     
  2. jcsd
  3. Oct 12, 2010 #2

    radou

    User Avatar
    Homework Helper

    If I remember right, I think I gave you a useful condition for a point to be in the closure of a set. Do you see how you can use it here?
     
  4. Oct 12, 2010 #3

    radou

    User Avatar
    Homework Helper

    Of course, your "steps" are a correct way to prove equivalence of statements, from a logical point of view.
     
  5. Oct 12, 2010 #4
    No I don't see how I can use it here in this problem.

    How would I start my step two? I know I assume alpha belongs to S closure, but I am not sure where to go from there.
     
  6. Oct 13, 2010 #5

    radou

    User Avatar
    Homework Helper

    A point x is in the closure of a set A if any neighbourhood of x intersects A. Now, what is an important property of the supremum (involving "ε")?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Prove alpha=sup(S) is equivalent to alpha belongs to S closure
  1. Prove inf(S)=-Sup(-S)? (Replies: 3)

  2. Prove V=S⊕S^(⊥) (Replies: 14)

Loading...