reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem Th34:
  (seq is divergent_to+infty or seq is divergent_to-infty) implies
  seq" is convergent & lim(seq")=0
proof
  assume
A1: seq is divergent_to+infty or seq is divergent_to-infty;
  now
    per cases by A1;
    suppose
A2:   seq is divergent_to+infty;
A3:   now
        let r be Real such that
A4:     0<r;
        consider n such that
A5:     for m st n<=m holds r"<seq.m by A2;
        take n;
        let m;
        assume n<=m;
        then
A6:     r"<seq.m by A5;
        then 1/seq.m<1/r" by A4,XREAL_1:76;
        then
A7:     1/seq.m<r by XCMPLX_1:216;
A8:     1/seq.m=(seq.m)" & (seq.m)"=(seq").m by VALUED_1:10,XCMPLX_1:215;
        0<r" by A4;
        hence |.(seq").m-0.|<r by A6,A7,A8,ABSVALUE:def 1;
      end;
      hence seq" is convergent;
      hence thesis by A3,SEQ_2:def 7;
    end;
    suppose
A9:   seq is divergent_to-infty;
A10:  now
        let r be Real such that
A11:    0<r;
A12:    -(r")<-0 by A11,XREAL_1:24;
        consider n such that
A13:    for m st n<=m holds seq.m<-(r") by A9;
        take n;
        let m;
        assume
A14:    n<=m;
        then seq.m<-(r") by A13;
        then 1/(-(r"))<1/seq.m by A12,XREAL_1:99;
        then ((-1)*(r"))"<1/seq.m by XCMPLX_1:215;
        then
A15:    (-1)"*(r"")<1/seq.m by XCMPLX_1:204;
        seq.m<-0 by A11,A13,A14;
        then 1/seq.m<0/seq.m by XREAL_1:75;
        then |.1/seq.m.|=-(1/seq.m) by ABSVALUE:def 1;
        then -(1*r)<-|.1/seq.m.| by A15;
        then |.1/seq.m.|<r by XREAL_1:24;
        then |.(seq.m)".|<r by XCMPLX_1:215;
        hence |.(seq").m-0.|<r by VALUED_1:10;
      end;
      hence seq" is convergent;
      hence thesis by A10,SEQ_2:def 7;
    end;
  end;
  hence thesis;
end;
