
theorem PXR2:
  for f be Function of NAT,REAL
  st f is negligible
  holds for r be Real st 0 < r
  holds
  ex N be Nat st for x be Nat st N <=x holds |. f.x .| < r
  proof
    let f be Function of NAT,REAL;
    assume AS: f is negligible;
    let r be Real;
    assume 0 < r;then
    consider c be non empty positive-yielding XFinSequence of REAL
    such that
    P1: for x be Nat holds (polynom(c)).x = 1/r by PXR1;
    consider N be Nat such that
    P2: for x be Nat
    st N <=x holds |. f.x .| < 1/((polynom(c)).x) by AS;
    take N;
    thus for x be Nat
    st N <=x holds |. f.x .| < r
    proof
      let x be Nat;
      assume N <=x;
      then
      |. f.x .| < 1/((polynom(c)).x) by P2;
      then
      |. f.x .| < 1/(1/r) by P1;
      hence
      |. f.x .| < r by XCMPLX_1:52;
    end;
  end;
