reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;
reserve X for non empty MetrSpace,
  x,y for Element of X,
  V for Subset of X,
  S,S1,T for sequence of X,
  Nseq for increasing sequence of NAT;

theorem Th8:
  S is bounded iff ex r,x st (0 < r & for n holds S.n in Ball(x,r))
proof
  thus S is bounded implies ex r,x st (0 < r & for n holds S.n in Ball(x,r))
  proof
    assume S is bounded;
    then consider r being Real, x such that
A1: 0 < r and
A2: rng S c= Ball(x,r);
    take q = r;
    take y = x;
    now
      let n;
      n in NAT by ORDINAL1:def 12;
      then S.n in rng S by FUNCT_2:4;
      hence S.n in Ball(y,q) by A2;
    end;
    hence thesis by A1;
  end;
  thus (ex r,x st (0 < r & for n holds S.n in Ball(x,r))) implies S is bounded
  proof
    given r,x such that
A3: 0 < r and
A4: for n holds S.n in Ball(x,r);
    reconsider r as Real;
    for x1 being object holds x1 in rng S implies x1 in Ball(x,r)
    proof
      let x1 be object;
      assume x1 in rng S;
      then consider x2 being object such that
A5:   x2 in dom S and
A6:   x1 = S.x2 by FUNCT_1:def 3;
      x2 in NAT by A5,FUNCT_2:def 1;
      hence thesis by A4,A6;
    end;
    then rng S c= Ball(x,r);
    hence thesis by A3;
  end;
end;
