1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Property of Compactness

  1. Jun 9, 2012 #1
    Suppose S[itex]\subset[/itex]ℝn is compact, f: S-->R is continous, and f(x)>0 for every x [itex]\in[/itex]S. Show that there is a number c>0 such that f(x) ≥ c for every x[itex]\in[/itex]S.



    Attempt:
    Since S is contained in Rn is compact, then S is closed and bounded.
    By the extreme value thm there exists values a,b that are an absolute minimum and absolute maximum respectively. Let c = f(a). Therefore by EVT f(c) ≤ f(x) in R.


    Well I have the solution manual and their solution is way different to what I attempted. Is there anything right about this?
     
  2. jcsd
  3. Jun 9, 2012 #2
    Your solution is correct. You might want to explain why c>0 however.
     
  4. Jun 9, 2012 #3
    Also, this should be c≤f(x).
     
  5. Jun 9, 2012 #4
    But isn't c [itex]\in[/itex] S. SO wouldn't the function map c to f(c)?



    Is the fact c having to be positive because if c < 0 ==> f(a) < f(c) ?
     
  6. Jun 9, 2012 #5
    You defined c=f(a). So c is an element of [itex]f(S)\subseteq \mathbb{R}[/itex]. Writing f(c) makes no sense.

    Again, f(c) makes no sense.
     
  7. Jun 9, 2012 #6
    got it now, thanks foe the help.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook