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

theorem Seq3:
  for n being Nat holds
    Sum (Reci-seq1, n, 1) < 2 / 3
  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;
W2: a.0 = 1 / (1 * 0 + - 1 / 2) by AlgDef
       .= -2;
V1: a.1 = 1 / (1 * 1 + - 1 / 2) by AlgDef
       .= 2;
V2: b.0 = -(((rseq (0,1,1, 1/2))).0) by VALUED_1:8
       .= - (1 / (1 * 0 + 1 / 2)) by AlgDef
       .= - 2;
V3: b.1 = -(((rseq (0,1,1, 1/2))).1) by VALUED_1:8
       .= - (1 / (1 * 1 + 1 / 2)) by AlgDef
       .= - 2 / 3;
    s.0 = (a.0 + b.0) by Tele2; then
V4: s.0 + s.1 = (a.0 + b.0) + (a.1 + b.1) by Tele2
             .= -2 + -2 / 3 by V1,V3,W2,V2;
V5: (Partial_Sums s).1 = (Partial_Sums s).0 + s.(0 + 1) by SERIES_1:def 1
       .= -2 + -2 / 3 by V4,SERIES_1:def 1;
W1: b.n = -(((rseq (0,1,1, 1/2))).n) by VALUED_1:8
       .= - (1 / (1 * n + 1 / 2)) by AlgDef
       .= - 1 / (n + 1 / 2);
W3: (Partial_Sums s).n = -2 + b.n by W2,ff,Telescoping,Tele2;
    b.n + 2 / 3 < 0 + 2 / 3 by XREAL_1:8,W1;
    hence thesis by W3,V5;
  end;
