reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem Th8:
  s is bounded iff ex r being Real st (0<r & for n holds |.s.n.|<r)
proof
  thus s is bounded implies
     ex r being Real st (0<r & for n holds |.s.n.|<r)
  proof
    assume s is bounded;
    then consider r being Real such that
A1: for n holds |.s.n.|<r;
    take r;
    now
      let n;
      0 <= |.s.n.| by COMPLEX1:46;
      hence 0< r by A1;
    end;
    hence thesis by A1;
  end;
  thus thesis;
end;
