reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th81:
  seq is bounded implies (lim_inf seq <= r iff for s st 0<s holds
  for n ex k st seq.(n+k)<r+s )
proof
  set seq1 = inferior_realsequence seq;
  assume
A1: seq is bounded;
  then
A2: seq1 is bounded by Th56;
  thus lim_inf seq <= r implies for s st 0<s holds for n ex k st seq.(n+k)<r+s
  proof
    assume
A3: lim_inf seq <= r;
    let s such that
A4: 0<s;
    for n ex k st seq.(n+k)<r+s
    proof
      let n;
      consider k such that
A5:   seq.(n+k) < seq1.n+s by A1,A4,Th40;
      seq1.n <= r by A2,A3,Th9;
      then seq.(n+k)+seq1.n < r+(seq1.n+s) by A5,XREAL_1:8;
      then seq.(n+k) < r+(seq1.n+s)-seq1.n by XREAL_1:20;
      hence thesis;
    end;
    hence thesis;
  end;
  assume
A6: for s st 0<s holds for n ex k st seq.(n+k)<r+s;
  for s st 0<s holds lim_inf seq<=r+s
  proof
    let s such that
A7: 0<s;
    for n holds seq1.n <= r+s
    proof
      let n;
      consider k such that
A8:   seq.(n+k)<r+s by A6,A7;
      seq1.n <= seq.(n+k) by A1,Th40;
      hence thesis by A8,XXREAL_0:2;
    end;
    hence thesis by Th9;
  end;
  hence thesis by XREAL_1:41;
end;
