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    Lipschitz functions

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    From Convergence, Open, and Closed Sets

    I am attaching my reference pdf file.
    Please prove #11 on page 55 based on my reference.

    Let M_X be the set of all functions f in C_[a,b] satisfying a Lipschitz condition.....

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    Solution Preview

    The explanations are in the attached text in two formats.

    Problem 11
    Metric space C_[a,b] is defined on page 39 as the set of all functions continuous on interval [a,b] with metric
    ρ(f,g)= max┬(a≤t≤b)⁡〖|f(t)-g(t)|〗 (0.1)
    (a.A) Suppose f(t) is a limit point for C_[a,b] so that there is a sequence of functions f_n→f converging to it. With metric defined by equation (0.1) it means simultaneous (uniform) convergence for each argument, f_n (t)→f(t), ∀t ∈[a,b]. To prove that f(t)∈ C_[a,b] we use triangle inequalities to construct inequality
    |f(t)-f(t^' )|≤|f(t)-f_n (t)|+|f_n (t)-f_n (t^' )|+|f_n (t^' )-f(t^' )| (a.1)
    to see that the condition of continuity can be satisfied for any t ∈[a,b] by taking t^'→t and using the fact that f_n∈C_[a,b] .
    (a.B) To prove that f(t) also satisfies the Lipschitz condition provided all f_n (t) do,
    |f_n (t)-f_n (t^' )|≤K|t-t^' | (a.2)
    we use ...

    Solution Summary

    This provides an example of a proof regarding functions that satisfy a Lipschitz condition.