reserve c, c1, c2, d, d1, d2, e, y for Real,
  k, n, m, N, n1, N0, N1, N2, N3, M for Element of NAT,
  x for set;

theorem Th5:
  Big_Oh(seq_logn) c= Big_Oh(seq_n^(1/2)) &
  not Big_Oh(seq_logn) = Big_Oh(seq_n^(1/2))
proof
  set g = seq_n^(1/2);
  set f = seq_logn;
A1: lim (f/"g) = 0 by Lm11;
A2: f/"g is convergent by Lm11;
  then not g in Big_Oh(f) by A1,ASYMPT_0:16;
  then
A3: not f in Big_Omega(g) by ASYMPT_0:19;
  f in Big_Oh(g) by A2,A1,ASYMPT_0:16;
  hence thesis by A3,Th4;
end;
