 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;

theorem Th26:
  for a,b being Real_Sequence st (for i being Nat holds a.i <= b.i) &
  a is non-decreasing & b is non-increasing holds
  a is convergent & b is convergent
  proof
    let a,b be Real_Sequence;
    assume that
A1: for i be Nat holds a.i <= b.i and
A2: a is non-decreasing and
A3: b is non-increasing;
    now
      take r = b.0 + 1;
      hereby
        let n be Nat;
        reconsider n0 = n as ExtReal;
        0 in NAT; then
A4:     0 in dom b by SEQ_1:1;
        n in NAT by ORDINAL1:def 12;
        then n0 in dom b by SEQ_1:1; then
A5:     b.n0 <= b.0 by A4,A3;
        a.n <= b.n by A1;
        then a.n <= b.0 by A5,XXREAL_0:2;
        then a.n + 0 < b.0 +1 by XREAL_1:8;
        hence a.n < r;
      end;
    end; then
A6: a is bounded_above by SEQ_2:def 3;
    now
      take r = a.0 - 1;
      hereby
        let n be Nat;
        reconsider n0 = n as ExtReal;
        0 in NAT; then
A7:     0 in dom a by SEQ_1:1;
        n in NAT by ORDINAL1:def 12; then
        n0 in dom a by SEQ_1:1; then
A8:     a.0 <= a.n0 by A7,A2;
        a.n <= b.n by A1;
        then a.0 <= b.n by A8,XXREAL_0:2;
        then a.0 - 1 < b.n - 0 by XREAL_1:15;
        hence r < b.n;
      end;
    end;
    then b is bounded_below by SEQ_2:def 4;
    hence thesis by A2,A3,A6;
  end;
