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prettymidget
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Homework Statement
Let property 1 be : If [ai,bi] is a sequence of intervals that are closed such that for each i the interval [a(i+1), b(i+1)] is either the left half of [ai,bi] or the right half, then there exists precisely 1 number in all intervals sequence.
Show if a field f satisfies this, then it satisfies the least upper bound property.
Homework Equations
The Attempt at a Solution
Let F be an ordered field that satisfies property 1. Let [ai,bi] be a sequence of intervals in F such that for each i the interval [a(i+1), b(i+1)] is either the left half of [ai,bi] or the right half. Then we have that there exists precisely 1 number in all intervals in this sequence.
The least upper bound property as my professor stated was that for any bounded above subset T of R, sup(T) exists.
I get intuitively why this has to follow from the bisection property thing, but I'm baffled on how to start the proof. My professor gave a hint in form of "How does the archimedean property follow from the bisection principle" but I'm not seeing it.
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