
theorem
  for F being sequence of ExtREAL st F is nonnegative & (ex n being
Element of NAT st (for k being Element of NAT st n <= k holds F.k = 0.) & (for
  k being Element of NAT st k <= n holds F.k <> +infty)) holds F is summable
proof
  let F be sequence of ExtREAL;
  assume
A1: F is nonnegative;
  given n being Element of NAT such that
A2: for k being Element of NAT st n <= k holds F.k = 0. and
A3: for k being Element of NAT st k <= n holds F.k <> +infty;
  for s being Element of NAT holds F.s in REAL
  proof
    let s be Element of NAT;
A4: s <= n implies F.s in REAL
    proof
A5:   F.s <> -infty by XXREAL_0:6,A1,Th38;
      assume s <= n; then
A6:   not F.s = +infty by A3;
      F.s in REAL or F.s in {-infty,+infty} by XBOOLE_0:def 3;
      hence thesis by A6,A5,TARSKI:def 2;
    end;
    n <= s implies F.s in REAL
    proof
      assume n <= s;
      then F.s = 0. by A2;
      hence thesis by XREAL_0:def 1;
    end;
    hence thesis by A4;
  end; then
  Ser(F).n in REAL by Th48;
  hence thesis by A1,A2,Th47;
end;
