reserve c, c1, d for Real,
  k for Nat,
  n, m, N, n1, N1, N2, N3, N4, N5, M for Element of NAT,
  x for set;

theorem :: Limit Rule for Big_Theta, Part 2 (page 88)
  for f,g being eventually-positive Real_Sequence st f/"g is convergent
  & lim( f/"g ) = 0 holds f in Big_Oh(g) & not f in Big_Theta(g)
proof
  let f,g be eventually-positive Real_Sequence;
  assume
A1: f/"g is convergent & lim( f/"g ) = 0;
  now
    assume f in Big_Theta(g);
    then f in Big_Omega(g) by XBOOLE_0:def 4;
    then g in Big_Oh(f) by Th19;
    hence contradiction by A1,Th16;
  end;
  hence thesis by A1,Th16;
end;
