reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th38:
  r>0 & (ex m st for n st n>=m holds |.s.n.|>=r) implies s is not
  convergent or lim s <> 0
proof
  assume
A1: r>0;
  given m such that
A2: for n st n>=m holds |.s.n.|>=r;
  now
    per cases;
    suppose
      s is not convergent;
      hence thesis;
    end;
    suppose
A3:   s is convergent;
      now
        assume lim s=0;
        then consider k such that
A4:     for n st n>=k holds |.s.n-0.|<r by A1,A3,SEQ_2:def 7;
        now
          let n such that
A5:       n>=m+k;
          m+k>=k by NAT_1:11;
          then n>=k by A5,XXREAL_0:2;
          then
A6:       |.s.n-0.|<r by A4;
          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;
