
theorem
  for rseq be Real_Sequence st (for n be Nat holds rseq.n=0)
  holds rseq is summable & Sum rseq = 0
proof
  let rseq be Real_Sequence such that
A1: for n be Nat holds rseq.n=0;
A2: for m be Nat holds Partial_Sums (rseq).m = 0
  proof
    defpred P[Nat] means rseq.$1 = (Partial_Sums rseq).$1;
    let m be Nat;
A3: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
A4:   rseq.k = (Partial_Sums (rseq)).k;
      thus rseq.(k+1) = 0 + (rseq).(k+1) .= rseq.k + rseq.(k+1) by A1
        .= (Partial_Sums rseq).(k+1) by A4,SERIES_1:def 1;
    end;
A5: P[0] by SERIES_1:def 1;
    for n be Nat holds P[n] from NAT_1:sch 2(A5,A3);
    hence (Partial_Sums rseq).m = rseq.m .= 0 by A1;
  end;
A6: for p be Real st 0<p ex n be Nat
   st for m being Nat st n<=m holds |.(Partial_Sums rseq).m-0 .|<p
  proof
    let p be Real such that
A7: 0<p;
    take 0;
    let m be Nat such that
    0<=m;
    |.(Partial_Sums rseq).m-0 .| = |.0-0 .| by A2
      .= 0 by ABSVALUE:def 1;
    hence thesis by A7;
  end;
  then
A8: Partial_Sums rseq is convergent by SEQ_2:def 6;
  then lim (Partial_Sums rseq) = 0 by A6,SEQ_2:def 7;
  hence thesis by A8,SERIES_1:def 2,def 3;
end;
