 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem
  for n being Nat holds
    (Partial_Sums Reci-seq1).n < -2
  proof
    let n be Nat;
    set s = Reci-seq1;
    set a = rseq (0,1,1, -1/2);
    set b = -(rseq (0,1,1, 1/2));
ff: for k being Nat holds b.k = -(a.(k+1))
    proof
      let k be Nat;
      b.k = - (((rseq (0,1,1, 1 / 2))).k) by VALUED_1:8
         .= - 1 / (1 * k + 1 / 2) by AlgDef
         .= - 1 / (1 * (k + 1) + - 1 / 2)
         .= - (a.(k + 1)) by AlgDef;
      hence thesis;
    end;
w3: a.0 = 1 / (1 * 0 + - 1 / 2) by AlgDef
       .= -2;
    b.n = -(((rseq (0,1,1, 1/2))).n) by VALUED_1:8
       .= - (1 / (1 * n + 1 / 2)) by AlgDef
       .= - 1 / (n + 1 / 2); then
    -2 + b.n < -2 + 0 by XREAL_1:8;
    hence thesis by w3,ff,Telescoping,Tele2;
  end;
