reserve n for Nat;

theorem Th39:
  for T being non empty TopSpace, F being SetSequence of the
  carrier of T, A being Subset of T st for i being Nat holds F.i = A holds
  Lim_inf F = Lim_sup F
proof
  let T be non empty TopSpace, F be SetSequence of the carrier of T, A be
  Subset of T;
  assume
A1: for i being Nat holds F.i = A;
  thus Lim_inf F c= Lim_sup F by Th31;
  thus Lim_sup F c= Lim_inf F
  proof
    let x be object;
    assume x in Lim_sup F;
    then ex G being subsequence of F st x in Lim_inf G by Def2;
    hence thesis by A1,Th12;
  end;
end;
