reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

theorem Th29:
  r > 0 & ( ex m st for n st n >= m holds ||.seq.n.|| >= r)
  implies not seq is convergent or lim seq <> 0.X
proof
  assume
A1: r > 0;
  given m such that
A2: for n st n >= m holds ||.seq.n.|| >= r;
  now
    per cases;
    suppose
      not seq is convergent;
      hence thesis;
    end;
    suppose
A3:   seq is convergent;
      now
        assume lim seq = 09(X);
        then consider k such that
A4:     for n st n >= k holds ||.seq.n - 09(X).|| < r by A1,A3,CLVECT_2:19;
        now
          let n;
          assume
A5:       n >= m+k;
          m+k >= k by NAT_1:11;
          then n >= k by A5,XXREAL_0:2;
          then ||.seq.n - 09(X).|| < r by A4;
          then
A6:       ||.seq.n.|| < r by RLVECT_1:13;
          m+k >= m by NAT_1:11;
          then n >= m by A5,XXREAL_0:2;
          hence contradiction by A2,A6;
        end;
        hence contradiction;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
