Question

In: Advanced Math

Suppose that a subset S of an ordered field F is not bounded above in F....

Suppose that a subset S of an ordered field F is not bounded above in F. Let T be a subset of F satisfying the property that, for each x ∈ S, there exists y ∈ T such that x ≤ y. Prove that T is not bounded above in F.

Solutions

Expert Solution

We will prove by contradiction that is not bounded above in   .

Suppose    is bounded above in   then there exist such that ,

for all   .

Let ,   be arbitrary .

By the given condition ,   as   .

Since   and   are subset of order field   and   ,  

Since    be arbitrary so   for all

Hence   is bounded above , a contradiction to   is not bounded above .

Hence   is not bounded above in   .

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Colby Messinger answered 2 years ago
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