Furthermore, non-satiation tells us that x*x**(pi) implies that x*>x**(pi) for all pI>p, and that<x*(pi implies that x*<x**(pi) for all pI <p,.Hence, since the set of p, for which x*<x**(P1) is bounded above there exists an upper bound p,g. And since the set of p, for which x*xx**(pi)is bounded below there exists a greatest lower bound plg. As it would be impossible to have x**(p1)>x*>x**(p1) it must be that p <pig IfP>PLB then there exists a continuum of values of pI at which x*≥x*(P1)andx*≤x*(p1) i.e. for which x*N x**(p1 but local non-satiation rules that out. so it must be that pla=pib=p, and by construction for all pi>p, and x*<x**(p1)andFurthermore, non-satiation tells us that x  x  p1  implies that x  xp1  for all p1   p1, and that x  x  p1  implies that x  xp1  for all p1   p1. Hence, since the set of p1for which x  xp1  is bounded above there exists an upper bound  p1,A. And since the set of p1for which x  xp1  is bounded below there exists a greatest lower bound  p1,B. As it would be impossible to have xp1   x  xp1 , it must be that  p1,A   p1,B. If  p1,A   p1,B then there exists a continuum of values of p1 at which x  xp1  and x  xp1 , i.e. for which x  xp1  but local non-satiation rules that out, so it must be that  p1,A   p1,B   p1 and by construction for all p1   p1, and x  xp1  and