
theorem
  for f be Function of NAT,REAL
  st f is negligible
  holds f is convergent & lim f = 0
  proof
    let f be Function of NAT,REAL;
    assume AS: f is negligible;
    A1:for p be Real st 0<p ex n be Nat st for m be Nat
    st n<=m holds |. f.m - 0 .| < p
    proof
      let p be Real;
      assume 0<p;
      then
      consider N be Nat such that
      P1: for x be Nat
      st N <=x holds |. f.x .| < p by AS,PXR2;
      reconsider N as Nat;
      take N;
      let m be Nat;
      assume P2: N<=m;
      reconsider x=m as Element of NAT by ORDINAL1:def 12;
      thus |. f.m - 0 .| < p by P1,P2;
    end;
    hence f is convergent by SEQ_2:def 6;
    hence lim f = 0 by A1,SEQ_2:def 7;
  end;
