reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;
reserve seq for ExtREAL_sequence;
reserve e1,e2 for ExtReal;
reserve rseq for Real_Sequence;

theorem Th11:
  seq is bounded implies seq is Real_Sequence
proof
  assume
A1: seq is bounded;
  then seq is bounded_below;
  then rng seq is bounded_below;
  then consider UL being Real such that
A2: UL is LowerBound of rng seq by XXREAL_2:def 9;
A3: UL in REAL by XREAL_0:def 1;
  seq is bounded_above by A1;
  then rng seq is bounded_above;
  then consider UB being Real such that
A4: UB is UpperBound of rng seq by XXREAL_2:def 10;
A5: UB in REAL by XREAL_0:def 1;
A6: now
    let x be object;
    assume x in NAT;
    then
A7: seq.x in rng seq by FUNCT_2:4;
    then
A8: seq.x <> -infty by A2,A3,XXREAL_0:12,XXREAL_2:def 2;
    seq.x <> +infty by A5,A4,A7,XXREAL_0:9,XXREAL_2:def 1;
    hence seq.x in REAL by A8,XXREAL_0:14;
  end;
  dom seq =NAT by FUNCT_2:def 1;
  hence thesis by A6,FUNCT_2:3;
end;
