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
  S is_convergent_in_metrspace_to x iff for r st 0 < r holds Ball(x,r)
  contains_almost_all_sequence S
proof
  thus S is_convergent_in_metrspace_to x implies for r st 0 < r holds Ball(x,r
  ) contains_almost_all_sequence S by Th15;
  thus (for r st 0 < r holds Ball(x,r) contains_almost_all_sequence S) implies
  S is_convergent_in_metrspace_to x
  proof
    assume for r st 0 < r holds Ball(x,r) contains_almost_all_sequence S;
    then for V st x in V & V in Family_open_set X holds V
    contains_almost_all_sequence S by Th16;
    hence thesis by Th17;
  end;
end;
