ECON 10A Final: ECON 1OA UCSC F04final
Document Summary
You have 60 minutes to answer all four questions. Prove by induction that for any n = f2; 3; :::g a set that contains n elements has n(n(cid:0)1) two elements. Assume the supremum property holds and prove that the in(cid:133)mum property must also hold. For- mally, assume that for any a (cid:18) r that is non-empty and bounded above sup a exists and sup a 2 r. Show that for any b (cid:18) r that is non-empty and bounded below inf b exists and inf b 2 r. Hint: in class, we proved the supremum property using the completeness axiom; you can try and adjust that proof to the in(cid:133)mum. Different hint: if you do not remember how to do that, use the assumption that the supremum property holds (our class proof did not, obviously). Let m be a set and de(cid:133)ne d : m (cid:2) m !
