Order Theory/Total orders

{{proposition|series order induced by total orders is total|Whenever $$A$$ is a totally ordered set

{{proposition|lexicographic order induced by total orders is total|Whenever $$A$$ is a well-ordered set and $$(S_\alpha, \le_\alpha)_{\alpha \in A}$$ are totally ordered sets, the lexicographic order on $$\prod_{\alpha \in A} S_\alpha$$ is total.}}

{{proof|Let any two elements $$(s_\alpha)_{\alpha \in A}$$ and $$(t_\alpha)_{\alpha \in A}$$ of $$\prod_{\alpha \in A} S_\alpha$$ be given. Then either $$(s_\alpha)_{\alpha \in A} = (t_\alpha)_{\alpha \in A}$$, or there exists a smallest $$\beta \in A$$ so that $$s_\beta \neq t_\beta$$. Since $$\le_\beta$$ is total, either $$s_\beta < t_\beta$$ or $$s_\beta > t_\beta$$, and thus either $$(s_\alpha)_{\alpha \in A} < (t_\alpha)_{\alpha \in A}$$ or $$(s_\alpha)_{\alpha \in A} > (t_\alpha)_{\alpha \in A}$$.}}

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