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 convergent implies lim seq = lim_sup seq & lim seq = lim_inf seq
proof
  reconsider r= lim seq as Real;
  assume
A1: seq is convergent;
  then
A2: upper_bound(inferior_realsequence seq) <=
lower_bound(superior_realsequence seq) by Th55;
A3: seq is bounded by A1;
  then
A4: superior_realsequence seq is bounded & inferior_realsequence seq is
  bounded by Th56;
A5: for p holds 0 < p implies r - p <= upper_bound inferior_realsequence seq &
 lower_bound
  superior_realsequence seq <= r + p
  proof
    let p;
    assume 0 < p;
    then consider k such that
A6: for m st k <= m holds |.seq.m-r.|<p by A1,SEQ_2:def 7;
A7: for m st k <= m holds r - p <= seq.m & seq.m <= r + p
    proof
      let m;
      assume k <= m;
      then |.seq.m - r.| < p by A6;
      hence thesis by Th1;
    end;
    then for m st k <= m holds r - p <= seq.m;
    then
A8: r - p <= (inferior_realsequence seq).k by A3,Th43;
    for m st k <= m holds seq.m <= r + p by A7;
    then
A9: (superior_realsequence seq).k <= r + p by A3,Th45;
    (inferior_realsequence seq).k <= upper_bound inferior_realsequence seq &
     lower_bound
    superior_realsequence seq <= (superior_realsequence seq).k by A4,Th7,Th8;
    hence thesis by A8,A9,XXREAL_0:2;
  end;
A10: for p holds 0 < p implies r <= upper_bound inferior_realsequence seq + p
  proof
    let p;
    assume 0 < p;
    then r - p <= upper_bound inferior_realsequence seq by A5;
    hence thesis by XREAL_1:20;
  end;
  then
A11: r <= upper_bound inferior_realsequence seq by XREAL_1:41;
  r <= upper_bound inferior_realsequence seq by A10,XREAL_1:41;
  then
A12: r <= lower_bound superior_realsequence seq by A2,XXREAL_0:2;
  for p holds 0 < p implies lower_bound superior_realsequence seq <=
   r + p by A5;
  then
A13: lower_bound superior_realsequence seq <= r by XREAL_1:41;
  then upper_bound inferior_realsequence seq <= r by A2,XXREAL_0:2;
  hence thesis by A13,A11,A12,XXREAL_0:1;
end;
