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 Th82:
  seq is bounded implies (r <= lim_inf seq iff for s st 0<s holds
  ex n st for k holds r-s<seq.(n+k) )
proof
  set seq1 = inferior_realsequence seq;
  assume
A1: seq is bounded;
  then
A2: seq1 is bounded by Th56;
  thus r <= lim_inf seq implies for s st 0<s holds ex n st for k holds r-s<seq
  .(n+k)
  proof
    assume
A3: r <= lim_inf seq;
    let s such that
A4: 0<s;
    ex n st for k holds r-s<seq.(n+k)
    proof
      consider n1 be Nat such that
A5:   seq1.n1 > upper_bound seq1-s by A2,A4,Th7;
      seq1.n1 + upper_bound seq1 > r + (upper_bound seq1-s)
      by A3,A5,XREAL_1:8;
      then seq1.n1 + upper_bound seq1 > r - s + upper_bound seq1;
      then
A6:   (seq1.n1 + upper_bound seq1) - upper_bound seq1 > r-s by XREAL_1:20;
      now
        let k;
        seq1.n1 <= seq.(n1+k) by A1,Th40;
        then (r-s) + seq1.n1 < seq.(n1+k) + seq1.n1 by A6,XREAL_1:8;
        then (r-s) + seq1.n1 - seq1.n1 < seq.(n1+k) by XREAL_1:19;
        hence r-s<seq.(n1+k);
      end;
      hence thesis;
    end;
    hence thesis;
  end;
  assume
A7: for s st 0<s holds ex n st for k holds r-s<seq.(n+k);
  for s st 0<s holds r-s<=lim_inf seq
  proof
    let s;
    assume 0<s;
    then consider n1 be Nat such that
A8: for k holds r-s<seq.(n1+k) by A7;
    for k holds r-s<=seq.(n1+k) by A8;
    then
A9: r-s <= seq1.n1 by A1,Th42;
    seq1.n1 <= upper_bound seq1 by A2,Th7;
    hence thesis by A9,XXREAL_0:2;
  end;
  hence thesis by XREAL_1:57;
end;
