reserve n for Nat;

theorem Th31:
  for T being non empty TopSpace, A being SetSequence of the
  carrier of T holds Lim_inf A c= Lim_sup A
proof
  let T be non empty TopSpace, A be SetSequence of the carrier of T;
  let x be object;
  assume
A1: x in Lim_inf A;
  ex K being subsequence of A st x in Lim_inf K
  proof
    reconsider B = A as subsequence of A by VALUED_0:19;
    take B;
    thus thesis by A1;
  end;
  hence thesis by Def2;
end;
