1. Feels So Good…Hurts So Bad
2. So, ?(�, v) = ? implies ?(�, u) = ?. Suppose ?(�, v) = ? and ? v > ? u for some ? u ? ?(�, u) and ? v ? ?(�, v);Let v * (.) := v(.) + u(m � ? ? u ) ? v(m � ? ? u ). As v * ? [v], we have A(., v * ) = A(., v), ?(., v * ) = ?(., v) and u(m � ? ? u ) = v * (m � ? ? u ). So, U (�, v * ) = v * (m � ? ? v ) < v * (m � ? ? u ) = u(m � ? ? u ) = U (�, u). Thus, � ? A(m � ? ? u , u) \ A(m � ? ? u , v * ) = A(m � ? ? u , u) \ A(m � ? ? u , v)
3. Suppose u(a) < u(b). As O has the upper bound property, there exists c ? O such that c ? a and c ? b. As u(c) ? u(b) > u(a), we have c > a;As u(m � ) ? U (�, u) ? u(x) and u(O) is an interval, U (�, u) = u(b) for some b ? O. Consequently, ? u := m � ? b ? ?(�, u)