
theorem
  for F1,F2 being sequence of ExtREAL st F1 is nonnegative & (for n
  being Element of NAT holds F1.n <= F2.n) holds (F2 is summable implies F1 is
  summable)
proof
  let F1,F2 be sequence of ExtREAL;
  assume F1 is nonnegative; then
A1: 0. <= Ser(F1).0 by Th39;
  Ser(F1).0 <= sup rng Ser F1 by FUNCT_2:4,XXREAL_2:4; then
A2: SUM(F1) <> -infty by A1,XXREAL_0:6;
  assume
A3: for n being Element of NAT holds F1.n <= F2.n;
  assume F2 is summable; then
A4: not SUM(F1) = +infty by A3,Th42,XXREAL_0:9;
  SUM(F1) in REAL or SUM(F1) in {-infty,+infty} by XBOOLE_0:def 3;
  hence thesis by A4,A2,TARSKI:def 2;
end;
