reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

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;
  per cases;
  suppose
    not seq is convergent;
    hence thesis;
  end;
  suppose
A3: seq is convergent;
    now
      assume lim seq = 09(X);
      then consider k being Nat such that
A4:   for n being Nat
        st n >= k holds ||.seq.n - 09(X).|| < r by A1,A3,BHSP_2:19;
        set n = m+k;
        m+k >= k by NAT_1:11;
        then n >= k;
        then ||.seq.n - 09(X).|| < r by A4;
        then
A5:     ||.seq.n.|| < r;
        m+k >= m by NAT_1:11;
        then n >= m;
        hence contradiction by A2,A5;
    end;
    hence thesis;
  end;
end;
