
theorem Th44:
  for F being sequence of ExtREAL st F is nonnegative holds
  (ex n being Element of NAT st F.n = +infty) implies SUM(F) = +infty
proof
  let F be sequence of ExtREAL;
  assume
A1: F is nonnegative;
  given n being Element of NAT such that
A2: F.n = +infty;
A3: (ex k being Nat st n = k + 1) implies SUM(F) = +infty
  proof
    given k being Nat such that
A4: n = k + 1;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    reconsider y = Ser(F).k as R_eal;
    reconsider x = Ser(F).(k + 1) as R_eal;
A5: Ser(F).(k + 1) = y + F.(k + 1) by Def11;
    Ser(F).k <> -infty by A1,Th39,XXREAL_0:6; then
A6: x = +infty by A2,A4,A5,XXREAL_3:def 2;
    then x is UpperBound of rng Ser(F) by XXREAL_2:41;
    hence thesis by A6,FUNCT_2:4,XXREAL_2:55;
  end;
  n = 0 implies SUM(F) = +infty
  proof
    reconsider x = Ser(F).0 as R_eal;
    assume n = 0; then
A7: Ser(F).0 = +infty by A2,Def11;
    then x is UpperBound of rng Ser(F) by XXREAL_2:41;
    hence thesis by A7,FUNCT_2:4,XXREAL_2:55;
  end;
  hence thesis by A3,NAT_1:6;
end;
