
theorem
  for F being sequence of ExtREAL, x being R_eal st
  (ex n being Element of NAT st x <= F.n) & F is nonnegative holds x <= SUM(F)
proof
  let F be sequence of ExtREAL;
  let x be R_eal;
  assume that
A1: ex n being Element of NAT st x <= F.n and
A2: F is nonnegative;
  consider n being Element of NAT such that
A3: x <= F.n by A1;
A4: Ser(F).n <= SUM(F) by FUNCT_2:4,XXREAL_2:4;
  per cases by NAT_1:6;
  suppose
    n = 0;
    then Ser(F).n = F.n by SUPINF_2:def 11;
    hence thesis by A3,A4,XXREAL_0:2;
  end;
  suppose
    ex k being Nat st n = k + 1;
    then consider k being Nat such that
A5: n = k + 1;
    reconsider k as Element of NAT by ORDINAL1:def 12;
A6: 0. <= Ser(F).k by A2,SUPINF_2:40;
    Ser(F).n = Ser(F).k + F.(k + 1) by A5,SUPINF_2:def 11;
    then 0. + x <= Ser(F).n by A3,A5,A6,XXREAL_3:36;
    then x <= Ser(F).n by XXREAL_3:4;
    hence thesis by A4,XXREAL_0:2;
  end;
end;
