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
  seq is bounded implies (r = lim_sup seq iff for s st 0<s holds (for n
  holds ex k st seq.(n+k)>r-s) & ex n st for k holds seq.(n+k)<r+s )
proof
  assume
A1: seq is bounded;
  hence
  r = lim_sup seq implies for s st 0<s holds (for n holds ex k st seq.(n+
  k)>r-s) & ex n st for k holds seq.(n+k)<r+s by Th84,Th85;
  assume
A2: for s st 0<s holds (for n holds ex k st seq.(n+k)>r-s) & ex n st for
  k holds seq.(n+k)<r+s;
  then for s st 0<s holds ex n st for k holds seq.(n+k)<r+s;
  then
A3: lim_sup seq <= r by A1,Th85;
  for s st 0<s holds for n holds ex k st seq.(n+k)>r-s by A2;
  then r <= lim_sup seq by A1,Th84;
  hence thesis by A3,XXREAL_0:1;
end;
