Question

Assume that (F,+,⋅) is a commutative field totally ordered. Given 3 elements a, b, c in...

Assume that (F,+,⋅) is a commutative field totally ordered.

Given 3 elements a, b, c in F, prove that if a<b, then a+c<b+c.

Prove that if 0<a<b, then b^{-1}<a^{-1}

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Answer #1

Let   is a commutative field totally ordered.

Given that,

Let   i.e for some   we can say that  

as  (Proved)

Given that   and   i.e. both   are positive.

so for some   we can say that  

Now,

as,   

(Proved)

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