reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

theorem
  (for n holds seq.n=g/(n+r)) implies seq is convergent & lim seq =0
proof
  assume
A2: for n holds seq.n=g/(n+r);
  reconsider r1 = r as Real;
  deffunc U(Nat) = 1/($1+r1);
  consider seq1 such that
A3: for n holds seq1.n=U(n) from SEQ_1:sch 1;
A4: now
    let n be Element of NAT;
    thus (g(#)seq1).n=g*(seq1.n) by SEQ_1:9
      .=g*(1/(n+r)) by A3
      .=g*(1*(n+r)")
      .=g/(n+r)
      .=seq.n by A2;
  end;
A5: g(#)seq1 is convergent by A3,Th28,SEQ_2:7;
  lim (g(#)seq1)=g*(lim seq1) by A3,Th28,SEQ_2:8
    .=g*(0 qua Nat) by A3,Th29
    .=0;
  hence thesis by A5,A4,FUNCT_2:63;
end;
