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  3. let (f, <) be an ordered field, let s be a none

Let (F, <) be an ordered field, let S be a nonempty subset of F, let...

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Advance Math

Let (F, <) be an ordered field, let S be a nonempty subset ofF, let c ? F, and for purposes of this problem let cS = {cx | x ?S}. (Do not use this notation outside this problem without definingwhat you mean by the notation.) Assume that c > 0.

  1. (i) Show that an element b ? F is an upper bound for S if andonly if cb is an upper bound for cS.

    (ii) Show that S has a least upper bound if and only if cS has aleast upper bound, and that (when these sets have least upperbounds), they are related by

    l.u.b.(cS) = c · l.u.b.(S).

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