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Continuous limited function, thus uniformly continuous

  1. Dec 10, 2008 #1
    1. The problem statement, all variables and given/known data
    suppose f : [0,infinity) -> R is continuous, and there is an L in R, s.t. f(x) -> L, as x -> infinity. Prove that f is uniformly continuous on [0,infinity).


    2. Relevant equations
    limit at xo: |x-xo| < delta then |f(x) -L| < epsilon

    continuous |x-x0| < delta then |f(x) - f(xo) | < epsilon

    uniformly continuous: |x-y| < delta then |f(x) - f(y)| < epsilon.


    3. The attempt at a solution
    Let A be a number s.t.
    if |x - y| < delta1 , |f(x) - L | < epsilon for all x in [a,infinity).
    b/c f is continuous then |x-xo| < delta1 |f(x) - f(y) | < epsilon.
    Thus f is uniformly continuous on [a,infinity).

    Then,
    b/c f is continuous it is uniformly continuous on any compact domain, namely [0,a].
    |x-y| < delta2 ...

    Then pick delta = min{delta1,delta2}, then the rest should follow.

    Does that seem sound?
     
  2. jcsd
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