
theorem Th6:
  for M being non empty MetrSpace,
      s being sequence of the carrier of TopSpaceMetr(M),
      x being Point of TopSpaceMetr(M) holds
    x in lim_f s
      iff
    for b being Element of Balls(x) ex i being Nat st
    for j being Nat st i <=j holds s.j in b
  proof
    let M be non empty MetrSpace,
    s be sequence of the carrier of TopSpaceMetr(M),
    x be Point of TopSpaceMetr(M);
    Balls(x) is basis of BOOL2F NeighborhoodSystem x by Th5;
    hence thesis by CARDFIL2:97;
  end;
