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Uniform continuity proof on bounded sets

  1. Mar 15, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S.


    2. Relevant equations
    Uniform continuity: For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)|

    3. The attempt at a solution

    Every time my book has covered a similar topic, it uses subsequences, which I'm a bit uncomfortable with. Is this following proof valid?

    Let f be uniformly continuous on a bounded set, S (1), Then:

    For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)| (2)

    so (3) |f(x)|<|f(y)| + e, for all x in S

    Therefore, it's bounded on S
     
  2. jcsd
  3. Mar 15, 2013 #2

    jbunniii

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    How did you get (3)? You don't know that ##|x-y| < d## for all ##x,y \in S##.

    This same question was just asked yesterday. Perhaps the hints given in that thread will help:

    https://www.physicsforums.com/showthread.php?t=678514
     
  4. Mar 15, 2013 #3

    LCKurtz

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    I thought that was just a typo until..
    Until I saw you "use" it. It isn't even a sentence. I think you need to work on understanding the definition of uniform continuity.
     
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