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Homework Help: Limit of bounded functions

  1. Oct 18, 2011 #1
    1. The problem statement, all variables and given/known data
    Let f and g be real-valued functions defined on A ⊆ R and let c ∈ R be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that limx→c g(x) = 0. Prove that limx→c f(x)g(x) = 0.

    2. Relevant equations

    3. The attempt at a solution
    This isn't a very hard question, but it has to be done with no assumptions from calculus (Analysis 1).
    Is it sufficient to say:
    since f is bounded by (c-δ, c+δ) for some δ>0,
    then limx→cf(x) = L is bounded by (f(c-δ), f(c+δ)),
    and since f(x0) is a real number, for any x0 in that interval,

    limx→c f(x)g(x) = limx→c f(x)limx→c g(x) = L * 0 = 0

    I'm just not too sure what would be a formal proof...
  2. jcsd
  3. Oct 18, 2011 #2


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    Homework Helper

    What do you mean with "if f is bounded by (a, b), then the limit is bounded by (f(a), f(b))"? Do you mean: if |f(x)| < a then |lim f(x)| < f(a)?
    Because that is not necessarily true, if it even makes sense.

    I think that the formal proof you are after uses the epsilon-delta definition, i.e.
    [tex]\forall_{\epsilon > 0} \exists_{\delta(\epsilon) > 0} : |x - c| < \delta \implies |f(x)g(x)| < \epsilon[/tex]
    Of course you already know that
    [tex]\forall_{\epsilon > 0} \exists_{\delta'(\epsilon) > 0} : |x - c| < \delta' \implies |g(x)| < \epsilon[/tex]
  4. Oct 18, 2011 #3
    Thank you, that makes sense
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