
theorem
  for a,b being Real,s being Real_Sequence, S being sequence of
  Closed-Interval-MSpace(a,b) st S=s & a<=b & s is non-increasing holds S is
  convergent
proof
  let a,b be Real,s be Real_Sequence, S be sequence of
  Closed-Interval-MSpace(a,b);
  assume that
A1: S=s and
A2: a<=b and
A3: s is non-increasing;
  for n being Nat holds s.n>a-1
  proof
    let n be Nat;
    a<a+1 by XREAL_1:29;
    then
A4: a-1<a by XREAL_1:19;
A5:   n in NAT by ORDINAL1:def 12;
    dom S=NAT by FUNCT_2:def 1;
    then
A6: s.n in rng S by A1,FUNCT_1:def 3,A5;
    the carrier of Closed-Interval-MSpace(a,b)=[.a,b.] by A2,TOPMETR:10;
    then a<=s.n by A6,XXREAL_1:1;
    hence thesis by A4,XXREAL_0:2;
  end;
  then s is bounded_below by SEQ_2:def 4;
  hence thesis by A1,A2,A3,Th8;
end;
