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;
