Let S be an ordered set. Let A ⊂ S be a nonempty finite subset. Then A is bounded. Furthermore, inf A exists and is in A and sup A exists and is in A. Hint: Use induction.
How do I use induction to prove that the infinum and supremum exist?
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Even though this is phrased in more generality, the answer here applies (as would any answer to that question): http://math.stackexchange.com/q/259893/ That answer shows specifically how to use induction to show that $\sup A$ exists and is in $A$, and it can be adapted to prove the same about the $\inf$. – Jonas Meyer Jan 28 '15 at 04:27
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See also http://math.stackexchange.com/q/548806/ – Jonas Meyer Jan 28 '15 at 04:32