
theorem Th38:
  for a be NAT-valued Real_Sequence, b be non trivial Nat st
  rng a c= b & a is eventually-non-zero holds
  Liouville_constant (a,b) is liouville
  proof
    let a be NAT-valued Real_Sequence, b be non trivial Nat;
    assume
A1: rng a c= b & a is eventually-non-zero;
    set x = Liouville_constant (a,b);
    for n be non zero Nat ex p be Integer, q be Nat st q > 1 &
    0 < |. x - p/q .| < 1/q|^n by Th37,A1;
    hence thesis by Def2;
end;
