
theorem
  for T being non empty TopStruct, S being sequence of T st S is
  not convergent holds Lim S = {}
proof
  let T be non empty TopStruct, S be sequence of T;
  assume
A1: S is not convergent;
  set x = the Element of Lim S;
  assume
A2: Lim S <> {};
  then x in Lim S;
  then reconsider x as Point of T;
  S is_convergent_to x by A2,FRECHET:def 5;
  hence contradiction by A1;
end;
