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Uniform continuity and bounded

  1. Nov 8, 2005 #1
    Prove that if f is uniformly continuous on a bounded set S then f is bounded on S.

    Our book says uniform continuity on an interval implies regular continuity on the interval, and in the previous chapter we proved that if a function is continuous on some closed interval then it is bounded. Is that how I should prove this, or am I missing something? It seems to easy this way and that usually means I am overlooking something.

  2. jcsd
  3. Nov 8, 2005 #2
    You're overlooking the fact that not all bounded sets are intervals.

    A possible way to prove the result, is to use that a uniformly continuous function maps Cauchy sequences to Cauchy sequences. If you assume that f is unbounded on S, this will lead to a contradiction.
  4. Nov 8, 2005 #3
    I was just getting ready to say just because S is bounded doesnt mean S is closed and I need S to be closed to go the easy way. I supposed f is uniformly continuous and unbounded then I said since S is bounded then there is a convergent sequence {Xn} in S by bolzano and since {Xn} converges then {Xn} is cauchy. Since f is uniformly continuous on S then f(Xn) is cauchy so f is bounded, contradiction
  5. Nov 8, 2005 #4
    That's not correct, but on the right path. You don't actually have a contradiction (unbounded sets can (and always do!) contain Cauchy sequences). You need to place more requirements on x_n.
  6. Nov 8, 2005 #5
    what kind of requirements do I need? Do I need to say something about Xn converging to an element in S? Do I need to use subsequences?
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