
theorem
  for F being sequence of  ExtREAL st F is nonnegative & SUM(F) < +infty
    holds for n being Element of NAT holds F.n in REAL
proof
  let F be sequence of ExtREAL;
  assume that
A1: F is nonnegative and
A2: SUM(F) < +infty;
  let n be Element of NAT;
  defpred P[Nat] means F.$1 <= Ser(F).$1;
A3: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume F.k <= Ser(F).k;
    reconsider x = Ser(F).k as R_eal;
    x + F.(k+1) = Ser(F).(k+1) by SUPINF_2:def 11;
    hence thesis by A1,SUPINF_2:40,XXREAL_3:39;
  end;
A4: P[0] by SUPINF_2:def 11;
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A3);
  then
A5: F.n <= Ser(F).n;
  Ser(F).n <= SUM(F) by FUNCT_2:4,XXREAL_2:4;
  then F.n <= SUM(F) by A5,XXREAL_0:2;
  then
A6: F.n < +infty by A2,XXREAL_0:2;
A7: 0 in REAL by XREAL_0:def 1;
  0. = 0 & 0. <= F.n by A1,SUPINF_2:39;
  hence thesis by A6,XXREAL_0:46,A7;
end;
