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_inf seq iff for s st 0<s holds (for n
  ex k st seq.(n+k)<r+s) & ex n st for k holds r-s<seq.(n+k) )
proof
  assume
A1: seq is bounded;
  hence
  r = lim_inf seq implies for s st 0<s holds (for n ex k st seq.(n+k)<r+s
  ) & ex n st for k holds r-s<seq.(n+k) by Th81,Th82;
  assume
A2: for s st 0<s holds (for n ex k st seq.(n+k)<r+s) & ex n st for k
  holds r-s<seq.(n+k);
  then for s st 0<s holds ex n st for k holds r-s<seq.(n+k);
  then
A3: r <= lim_inf seq by A1,Th82;
  for s st 0<s holds for n ex k st seq.(n+k)<r+s by A2;
  then lim_inf seq <= r by A1,Th81;
  hence thesis by A3,XXREAL_0:1;
end;
