
theorem
  for m being Nat st 1 < m holds
    Liouville_seq (seq_const 1, m) is negligible
  proof
    let m be Nat;
    assume
AS: 1 < m;
    ex f being Function of NAT,REAL
      st for x be Nat holds f.x = 1 / (2 to_power x)
    proof
      deffunc F(Nat) = 1 / (2 to_power ($1));
      (ex f being Real_Sequence st for n being Nat holds f.n = F(n)) &
      for f1, f2 being Real_Sequence st
       (for n being Nat holds f1.n = F(n)) &
       (for n being Nat holds f2.n = F(n)) holds f1 = f2
         from IRRAT_1:sch 1;
      hence thesis;
    end; then
    consider f being Function of NAT,REAL such that
ACF: for x be Nat holds f.x = 1 / (2 to_power x);
   set g = Liouville_seq (seq_const 1, m);
   for x be Nat holds |. g.x .| <= |. f.x .|
   proof
     let x be Nat;
m1:  f.x = 1 / (2 to_power x) by ACF; then
M1:  g.x <= f.x by TLLC,AS;
     0 <= g.x
     proof
       per cases;
       suppose x = 0;
         hence thesis by DefLio;
       end;
       suppose x is non zero; then
         g.x = ((seq_const 1).x) / (m to_power (x!)) by DefLio;
         hence thesis;
       end;
     end; then
     -(f.x) <= g.x by m1; then
M3:  |. g.x .| <= f.x by M1,ABSVALUE:5;
     f.x <= |. f.x .| by ABSVALUE:4;
     hence thesis by XXREAL_0:2,M3;
    end;
    hence thesis by ACF,ASYMPT_3:25,ASYMPT_3:26;
  end;
