reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;
reserve seq for ExtREAL_sequence;
reserve e1,e2 for ExtReal;
reserve rseq for Real_Sequence;

theorem
  for seq be ExtREAL_sequence st seq is convergent holds lim seq =
  lim_inf seq & lim seq = lim_sup seq
proof
  let seq be ExtREAL_sequence;
  set a = lim_inf seq;
  assume seq is convergent;
  then
A1: lim_inf seq = lim_sup seq by Th40;
  per cases by XXREAL_0:14;
  suppose
    a in REAL;
    hence thesis by A1,Lm7;
  end;
  suppose
    a = +infty;
    hence thesis by A1,Lm5;
  end;
  suppose
    a = -infty;
    hence thesis by A1,Lm6;
  end;
end;
