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

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

(i) Show that an element b ∈ F is an upper bound for S if and only if cb is an upper bound for cS.

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

l.u.b.(cS) = c · l.u.b.(S).

Expert Solution

For (i), firstly, some of the basic properties of an ordered field have been proved. Then, the main result has been shown.

For (ii), the result in (i) has been used and the main result has been proved using the definition of the least upper bound or supremum of any subset of an ordered set.

Colby Messinger answered 1 year ago

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Question

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

Solutions

Expert Solution

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