
theorem
for seq be nonnegative ExtREAL_sequence
 st not Partial_Sums seq is convergent_to_+infty
  holds for n be Nat holds seq.n is Real
proof
   let seq be nonnegative ExtREAL_sequence;
   assume A2:not Partial_Sums seq is convergent_to_+infty;
   given N be Nat such that
A3: not seq.N is Real;
   not seq.N in REAL by A3; then
A4:seq.N = +infty or seq.N = -infty by XXREAL_0:14;
A6:Partial_Sums seq is non-decreasing by MESFUNC9:16;
   now let g be Real;
    assume 0 < g;
    take N;
    hereby let m be Nat;
     assume A7: N<=m;
     per cases;
     suppose N = 0; then
      (Partial_Sums seq).N = seq.N by MESFUNC9:def 1; then
A9:   g <= (Partial_Sums seq).N by A4,SUPINF_2:51,XXREAL_0:3;
      (Partial_Sums seq).N <= (Partial_Sums seq).m
        by A7,MESFUNC9:16,RINFSUP2:7;
      hence g <= (Partial_Sums seq).m by A9,XXREAL_0:2;
     end;
     suppose N <> 0; then
      consider N1 be Nat such that
A11:   N = N1 + 1 by NAT_1:6;
A12:  (Partial_Sums seq).N1 >= 0 by SUPINF_2:51;
      (Partial_Sums seq).N
        = (Partial_Sums seq).N1 + seq.N by A11,MESFUNC9:def 1
       .= +infty by A4,SUPINF_2:51,XXREAL_0:4,A12,XXREAL_3:39; then
      (Partial_Sums seq).m = +infty by A6,A7,RINFSUP2:7,XXREAL_0:4;
      hence g <= (Partial_Sums seq).m by XXREAL_0:3;
     end;
    end;
   end;
   hence contradiction by A2,MESFUNC5:def 9;
end;
