reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem
  (for n holds s.n>0) & (ex m st for n st n>=m holds s.(n+1)/s.n >= 1)
  implies s is not summable
proof
  assume that
A1: for n holds s.n>0 and
A2: ex m st for n st n>=m holds s.(n+1)/s.n >= 1;
  consider m such that
A3: for n st n>=m holds s.(n+1)/s.n>=1 by A2;
  defpred X[Nat] means s.(m+$1)>=s.m;
A4: for k st X[k] holds X[k+1]
  proof
    let k such that
A5: s.(m+k)>=s.m;
    s.(m+k)>0 & s.(m+k+1)/s.(m+k)>=1 by A1,A3,NAT_1:11;
    then s.(m+k+1)>=s.(m+k) by XREAL_1:191;
    hence thesis by A5,XXREAL_0:2;
  end;
A6: X[0];
A7: for k holds X[k] from NAT_1:sch 2(A6,A4);
A8: for k ex n st n>=k & not |.s.n - 0.|<s.m
  proof
    let k;
    take n = m + k;
    s.n>=s.m by A7;
    hence thesis by NAT_1:11,SEQ_2:1;
  end;
  s.m>0 by A1;
  then not lim s = 0 or s is not convergent by A8,SEQ_2:def 7;
  hence thesis by Th4;
end;
