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If you want to prove that 'complete plus densely ordered' implies connected you are almost forced to use the standard'standard' proof. For the real line you could also use the bisection method: if $I$ is convex and the union of two closed sets $A$ and $B$ take $a\in A$ and $b\in B$, with $a < b$, say. Now create two sequences $(a_n)_n$ (increasing) and $(b_n)_n$ (decreasing) with $a_n\in A$, $b_n\in B$ and $b_n-a_n=(b-a)2^{-n}$; then the common limit of these sequences belongs to $I\cap A\cap B$.
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If you want to prove that complete plus densely ordered' implies connected you are almost forced to use thestandard' proof. For the real line you could also use the bisection method: if $I$ is convex and the union of two closed sets $A$ and $B$ take $a\in A$ and $b\in B$, with $a < b$, say. Now create two sequences $(a_n)_n$ (increasing) and $(b_n)_n$ (decreasing) with $a_n\in A$, $b_n\in B$ and $b_n-a_n=(b-a)2^{-n}$; then the common limit of these sequences belongs to $I\cap A\cap B$.