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Math Help - Help understanding a proof (Uniform Convergence)

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    Help understanding a proof (Uniform Convergence)

    The lecturer proved a theorem in class, however I have found a step that he did redundant. But maybe I don't understand what he tried to do there. I uploaded the file.
    (The subject is Uniform convergece)

    Appreciate if someone can tell me why he did what he did.Thanks!!
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