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 Th17:
  (for V st x in V & V in Family_open_set X holds V
  contains_almost_all_sequence S) implies S is_convergent_in_metrspace_to x
proof
A1: for r st 0 < r holds x in Ball(x,r)
  proof
    let r;
    assume 0 < r;
    then dist(x,x) < r by METRIC_1:1;
    hence thesis by METRIC_1:11;
  end;
  assume
A2: for V st x in V & V in Family_open_set X holds V
  contains_almost_all_sequence S;
  for r st 0 < r ex m st for n st m <= n holds dist(S.n,x) < r
  proof
    let r;
    assume 0 < r;
    then x in Ball(x,r) by A1;
    then Ball(x,r) contains_almost_all_sequence S by A2,PCOMPS_1:29;
    then consider m1 such that
A3: for n st m1 <= n holds S.n in Ball(x,r);
    take k = m1;
    let n;
    assume k <= n;
    then S.n in Ball(x,r) by A3;
    hence thesis by METRIC_1:11;
  end;
  hence thesis;
end;
