# Bounded intervals in R and bisection method proof

#### prettymidget

1. The problem statement, all variables and given/known data

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.
2. Relevant equations

3. 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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