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 Th84:
  seq is bounded implies (r <= lim_sup seq iff for s st 0<s holds
  for n ex k st seq.(n+k)>r-s )
proof
  set seq1 = superior_realsequence seq;
  assume
A1: seq is bounded;
  then
A2: seq1 is bounded by Th56;
  thus r <= lim_sup seq implies for s st 0<s holds for n ex k st seq.(n+k)>r-s
  proof
    assume
A3: r <= lim_sup seq;
    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,Th41;
      seq1.n >= r by A2,A3,Th10;
      then seq1.n+seq.(n+k) > r+(seq1.n-s) by A5,XREAL_1:8;
      then seq.(n+k) > r+seq1.n-s-seq1.n by XREAL_1:19;
      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_sup seq>=r-s
  proof
    let s such that
A7: 0<s;
    for n holds r-s <= seq1.n
    proof
      let n;
      consider k such that
A8:   r-s<seq.(n+k) by A6,A7;
      seq.(n+k) <= seq1.n by A1,Th41;
      hence thesis by A8,XXREAL_0:2;
    end;
    hence thesis by Th10;
  end;
  hence thesis by XREAL_1:57;
end;
