reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th44:
  |.z.| < 1 & (for n holds seq.n=z |^ (n+1)) implies seq is
  convergent & lim(seq)=0c
proof
  assume that
A1: |.z.| <1 and
A2: for n holds seq.n=z |^ (n+1);
A3: now
    let n;
    thus seq.(n+1) = z |^ ((n+1)+1) by A2
      .= z |^ (n+1) * z by NEWTON:6
      .= seq.n * z by A2;
  end;
A4: now
    assume |.z.| = 0;
    then
A5: z =0c by COMPLEX1:45;
A6: now
      let n;
      thus seq.n = 0c |^ (n+1) by A2,A5
        .=(0c GeoSeq).n * 0c by Def1
        .=0c;
    end;
    hence seq is convergent by COMSEQ_2:9;
    thus thesis by A6,COMSEQ_2:9,10;
  end;
A7: seq.0= z |^ (0+1) by A2
    .=z;
  now
A8: 0 <= |.z.| by COMPLEX1:46;
    assume |.z.| <> 0;
    hence thesis by A1,A7,A3,A8,Th43;
  end;
  hence thesis by A4;
end;
