There are several ways to show that. The direct way is just to use definition of sup and inf.

1) We'd like to show that

inf(a, b)=b, (a<b).

If a is a minimal element from (a, b), then by the definition of inf this means that ∀x∈ (a, b)

i)x≥a

ii) ∀ε>0 ∃x' ∈ (a, b) such that

a+ε<x'.

Now, x was the minimal element but we found another one x'-ε, which is smaller then x. This proves that a inf(a,b). Similar one can show that a=inf[a,b].

As for the sup, you can use the fact sup{-x}=-inf{x}, where {-x} is the set elements, opposite to x sign.

Bobosharif S.

tutor

Most probably it works. I don't have a counterexample.

But I'm not sure what you mean by ordered field? If you specify definition or ordered filed, then I might tell you more.

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01/31/18

Morgan I.

01/31/18