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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 Summary

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

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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 ...

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