reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th68:
  ex s being Rational_Sequence st s is convergent & lim s = a &
  for n holds s.n>=a
proof
  deffunc O(Nat) = [/($1+1)*a\]/($1+1);
  consider s being Real_Sequence such that
A1: for n holds s.n = O(n) from SEQ_1:sch 1;
  rng s c= RAT
  proof
    let y be object;
    assume y in rng s;
    then consider n being Element of NAT such that
A2: s.n = y by FUNCT_2:113;
    s.n = O(n) by A1;
    hence thesis by A2,RAT_1:def 2;
  end;
  then reconsider s as Rational_Sequence by RELAT_1:def 19;
  deffunc O(Nat) = 1/($1+1);
  consider s2 being Real_Sequence such that
A3: for n holds s2.n = O(n) from SEQ_1:sch 1;
  take s;
  set s1 = seq_const a;
  set s3 = s1 + s2;
A4: s2 is convergent by A3,SEQ_4:31;
  then
A5: s3 is convergent;
A6: now
    let n;
    (n+1)*a + 1 >= [/(n+1)*a\] by INT_1:def 7;
    then ((n+1)*a + 1)*(n+1)" >= [/(n+1)*a\]/(n+1) by XREAL_1:64;
    then (a/(n+1))*(n+1) + 1/(n+1) >= s.n by A1;
    then a + 1/(n+1) >= s.n by XCMPLX_1:87;
    then s1.n + 1/(n+1) >= s.n by SEQ_1:57;
    then
A7: s1.n + s2.n >= s.n by A3;
    [/(n+1)*a\] >= (n+1)*a by INT_1:def 7;
    then [/(n+1)*a\]*(n+1)" >= a*(n+1)*(n+1)" by XREAL_1:64;
    then [/(n+1)*a\]*(n+1)" >= a*((n+1)*(n+1)");
    then [/(n+1)*a\]*(n+1)" >= a*1 by XCMPLX_0:def 7;
    then [/(n+1)*a\]/(n+1) >= s1.n by SEQ_1:57;
    hence s1.n <= s.n & s.n <= s3.n by A1,A7,SEQ_1:7;
  end;
  lim s2 = 0 by A3,SEQ_4:31;
  then
A8: lim s3 = s1.0 + 0 by A4,SEQ_4:42
    .= a by SEQ_1:57;
A9: lim s1 = s1.0 by SEQ_4:26
    .= a by SEQ_1:57;
  hence s is convergent by A5,A8,A6,SEQ_2:19;
  thus lim s = a by A5,A8,A9,A6,SEQ_2:20;
  let n;
  s.n>=s1.n by A6;
  hence thesis by SEQ_1:57;
end;
