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 holds lim_inf seq <= lim_sup seq
proof
  let seq be ExtREAL_sequence;
A1: lim superior_realsequence seq = lim_sup seq by Th36;
A2: inferior_realsequence seq is convergent by Th37;
A3: now
    let n be Nat;
A4: seq.n <= (superior_realsequence seq).n by Th8;
    (inferior_realsequence seq).n <=seq.n by Th8;
    hence (inferior_realsequence seq).n <= (superior_realsequence seq).n by A4,
XXREAL_0:2;
  end;
A5: lim inferior_realsequence seq = lim_inf seq by Th37;
  superior_realsequence seq is convergent by Th36;
  hence thesis by A1,A2,A5,A3,Th38;
end;
