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Uniform Convergence and the Uniform Metric

  1. Dec 25, 2012 #1
    Let X be a set, and let fn : X---> R be a sequence of functions. Let ρ be the uniform metric on the space RX. Show that the sequence (fn) converges uniformly to the function f:X--> R if and only if the sequence (fn) converges to f as elements of the metric space (RX, ρ). [Note: the ρ's should have a bar over them.]

    I have a question concerning one direction of our implication.

    [My attempt at the solution.]

    Since I do not know how to make a d with a bar over it, I'll use ∂ to denote the standard bounded metric relative to d.

    Proof:

    Assume the sequence (fn) converges to f as elements of the metric space (RX, ρ).

    Then, for any ε>0, there is a positive integer N such that ρ(fn,f)<ε for all n≥N.

    That is, for n ≥ N, the sup{∂(fn(x),f(x))| x in X} < ε.

    If ε≤1, our metrics are the same.

    Hence for ε≤1, there exisits an integer N such that for n ≥ N, the sup{∂(fn(x),f(x))| x in X} < ε.

    But since ∂(fn(x),f(x)) = d(fn(x),f(x)) for n ≥ N, d(fn(x),f(x)) ≤ sup{∂(fn(x),f(x))| x in X} < ε for all n≥N and x in X, which satisfies our condition for uniform convergence.



    My question is about ε. Is this the right approach?
     
  2. jcsd
  3. Dec 25, 2012 #2

    jbunniii

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    To display a bar, you can use the TeX command \bar. For example, to typeset [itex]\bar{\rho}[/itex], use \bar{\rho}.

    Just to make sure I understand the question correctly, let me paraphrase it with proper typesetting: by standard bounded metric relative to [itex]d[/itex], you mean
    [tex]\bar{d}(x,y) = \min\{d(x,y), 1\}[/tex]
    and that the following definitions are in use
    [tex]\rho(f,g) = \sup\{d(f(x),g(x)) : x \in X\}[/tex]
    [tex]\bar{\rho}(f,g) = \sup\{\bar{d}(f(x),g(x)) : x \in X\}[/tex]
    and that the goal is to show that
    [tex]\bar{\rho}(f_n,f) \rightarrow 0 \implies \rho(f_n,f) \rightarrow 0[/tex]
    Assuming I interpreted the question correctly, what you have done is fine for [itex]\epsilon \leq 1[/itex]. For [itex]\epsilon > 1[/itex], you can simply use the [itex]N[/itex] that works for [itex]\epsilon = 1[/itex].
     
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