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 Th85:
  seq is bounded implies (lim_sup seq <= r iff for s st 0<s holds
  ex n st for k holds seq.(n+k)<r+s )
proof
  set seq1 = superior_realsequence seq;
  assume
A1: seq is bounded;
  then
A2: seq1 is bounded by Th56;
  thus lim_sup seq <= r implies for s st 0<s holds ex n st for k holds seq.(n+
  k)<r+s
  proof
    assume
A3: lim_sup seq <= r;
    for s st 0<s holds ex n st for k holds seq.(n+k)<r+s
    proof
      let s such that
A4:   0<s;
      ex n st for k holds seq.(n+k)<r+s
      proof
        consider n1 be Nat such that
A5:     seq1.n1 < lower_bound seq1+s by A2,A4,Th8;
        seq1.n1 + lower_bound seq1 < r + (lower_bound seq1+s)
        by A3,A5,XREAL_1:8;
        then seq1.n1 + lower_bound seq1 < r + s + lower_bound seq1;
        then
A6:     (seq1.n1 + lower_bound seq1) - lower_bound seq1 < r+s by XREAL_1:19;
        now
          let k;
          seq.(n1+k) <= seq1.n1 by A1,Th41;
          then seq.(n1+k) + seq1.n1 < (r+s) + seq1.n1 by A6,XREAL_1:8;
          then seq.(n1+k) + seq1.n1 - seq1.n1 < (r+s) by XREAL_1:19;
          hence seq.(n1+k) < r+s;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
  assume
A7: for s st 0<s holds ex n st for k holds seq.(n+k)<r+s;
  for s st 0<s holds lim_sup seq<=r+s
  proof
    let s;
    assume 0<s;
    then consider n1 be Nat such that
A8: for k holds seq.(n1+k)<r+s by A7;
    for k holds seq.(n1+k)<=r+s by A8;
    then
A9: seq1.n1<=r+s by A1,Th44;
    lower_bound seq1 <= seq1.n1 by A2,Th8;
    hence thesis by A9,XXREAL_0:2;
  end;
  hence thesis by XREAL_1:41;
end;
