<p>How to prove that If E⊂ R is non-empty, then sup E
exists in R?</p>
<p>If R is referring to the set of reals, then sup E doesn’t necessarily exist (e.g. let E be the set of all integers, then sup E does not exist).</p>
<p>Otherwise, if R is bounded, sup E definitely exists since sup E ≤ sup R and if E has a supremum, it is unique, but sup E is not necessarily in R. For example, let R be the set of irrationals in (-2,2), and E be the set of irrationals in (-1,1). Then sup E = 1 which is not in R.</p>
<p>I think this question was posted in the wrong forum</p>
<p>Maybe, I have a slight suspicion that SAT doesn’t test anything on sup/inf. Also I still don’t think the OP’s post is correct mathematically.</p>
<p>Maybe it was posted by a Super Mum wanting to test our small infimum. </p>