a condition equivalent to compactness in linearly ordered spaces

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Does anyone know where can I find a proof to this proposition: A linearly ordered topological space is compact if and only if every bounded subset has an infimum and a supremum.

Thank you,

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See ProofWiki's proof of a slight generalization. Note: any errors in that proof are probably my own.

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An "inductive proof" is given in this short note.