Infimum and supremum of finite ordered subsets I am currently taking an introductory proofs course, and I have come across this problem.  It's asking to prove the following:


  Let $S$ be an ordered set. Let $A$ be a non-empty 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.
It's finite, so it's got $n$ elements, say. If $n=1$, you're done (why?).

If $n &gt; 1$, then try removing some element $a$. The resulting set (by inductive hypothesis) has an inf and a sup; now put the element $a$ back in.
It's finite, so it's got $n$ elements, say. If $n=1$, you're done (why?).

If $n &gt; 1$, then try removing some element $a$. The resulting set (by inductive hypothesis) has an inf and a sup; now put the element $a$ back in.