I've already asked a similar question yesterday (is there a well-order such that $X$ has a largest element). Can one prove the existence of a well-order on an infinite set $X$ such that $\forall x \in X \exists y \in X: x \leq y$?
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1Every ordering trivially satisfies your condition. Did you mean $x<y$? – freakish Jan 10 '19 at 15:34
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Yes. Every linear order for a finite set has a maximum. – William Elliot Jan 11 '19 at 03:12