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Let $F$ be an ordered field and $[a,b]\subset F$ be an non-degenerate closed bound interval in $F$. If $P,Q$ is a partition of $[a,b]$ with $a\in P$ and $b\in Q$ Show that there is a family of closed bounded intervals $\{[a_k,b_k\}_{k\mathbb{N}}\subset [a,b]$ such that for each $k\in\mathbb{N}$:

(i) $a_k\in P$ and $b_k\in Q$

(ii) $[a_{k+1}.b_{k+1}]\subset[a_k,b_k]$

(iii) $b_k-a_k=\frac{b-a}{2^{k-1}}$

I tried to use an induction argument for this by choosing $a_1=a\in P$ so then $b_1=b\in Q$ so then by the induction hypothesis we have $[a_k,b_k]\cap P\neq \emptyset$ so we can pick an element from the intersection to be $a_{k+1}$ and then define $b_{k+1}=a_{k+1}+\frac{b-a}{2^k}$. Then it is trivial to see that (ii) and (iii) are true for $k+1$, however I am not able to see why $b_{k+1}\in Q$, so any help to show this is greatly appreciated and needed.

Thanks in advance.

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Given $a_k\in P$, $b_k\in Q$ with $b_k-a_k=\frac{b-a}{2^k}$, let $c_k=\frac{a_k+b_k}{2}$. Then if $c_k\in P$, let $a_{k+1}=c_k$, $b_{k+1}=b_k$; otherwise we have $c_k\in Q$ and can let $a_{k+1}=a_k$, $b_{kj+1}=c_k$.

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