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 Th33:
  for seq be ExtREAL_sequence st (for n be Element of NAT holds
  seq.n <= -infty) holds seq is convergent_to_-infty
proof
  let seq be ExtREAL_sequence;
  assume
A1: for n be Element of NAT holds seq.n <= -infty;
  now
    let g be Real;
    assume g < 0;
    now
      let m be Nat;
      assume 0 <= m;
      m is Element of NAT by ORDINAL1:def 12;
      then
A2:   seq.m <= -infty by A1;
      -infty <= g by XXREAL_0:5;
      hence seq.m <= g by A2,XXREAL_0:2;
    end;
    hence ex n be Nat st for m be Nat st n <=m holds seq.m <= g;
  end;
  hence thesis by MESFUNC5:def 10;
end;
