theorem
  0<=r & (for n holds seq.n=g/(n*n+r)) implies seq is convergent & lim seq=0
proof
  assume that
A1: 0<=r and
A2: for n holds seq.n=g/(n*n+r);
  reconsider r1 = r as Real;
  deffunc U(Nat) = 1/($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*n+r)) by A3
      .=g*(1*(n*n+r)")
      .=g/(n*n+r)
      .=seq.n by A2;
  end;
A5: g(#)seq1 is convergent by A1,A3,Th32,SEQ_2:7;
  lim (g(#)seq1)=g*(lim seq1) by A1,A3,Th32,SEQ_2:8
    .=g*(0 qua Nat) by A1,A3,Th33
    .=0;
  hence thesis by A5,A4,FUNCT_2:63;
end;
