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 & s1.n = n-root (s.n)) & s1 is convergent & lim s1
  > 1 implies s is not summable
proof
  assume that
A1: for n holds s.n>=0 & s1.n = n-root (s.n) and
A2: s1 is convergent and
A3: lim s1 > 1;
  set r = lim s1 - 1;
  r > 0 by A3,XREAL_1:50;
  then consider m such that
A4: for n st m<=n holds |.s1.n - lim s1.| < r by A2,SEQ_2:def 7;
  for n st m<=n holds s1.n >= 1
  proof
    let n;
    assume m<=n;
    then |.s1.n - lim s1.| < r by A4;
    then s1.n - lim s1 > - r by SEQ_2:1;
    then s1.n - lim s1 + lim s1 > - r + lim s1 by XREAL_1:6;
    hence thesis;
  end;
  hence thesis by A1,Th29;
end;
