reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem
  (for n holds seq.n <> 0.X) & (for n holds Rseq.n = ||.seq.(n+1).||/||.
  seq.n.||) & Rseq is convergent & lim Rseq > 1 implies not seq is summable
proof
  assume that
A1: for n holds seq.n <> 09(X) and
A2: for n holds Rseq.n = ||.seq.(n+1).||/||.seq.n.|| and
A3: Rseq is convergent and
A4: lim Rseq > 1;
  lim Rseq - 1 > 0 by A4,XREAL_1:50;
  then consider m being Nat such that
A5: for n being Nat st n >= m
    holds |.Rseq.n - lim Rseq.| < lim Rseq - 1 by A3,
SEQ_2:def 7;
  now
    let n;
A6: m + 1 >= m by NAT_1:11;
A7: Rseq.n = ||.seq.(n+1).||/||.seq.n.|| by A2;
    assume n >= m + 1;
    then n >= m by A6,XXREAL_0:2;
    then
    |.||.seq.(n+1).||/||.seq.n.|| - lim Rseq.| < lim Rseq - 1 by A5,A7;
    then - (lim Rseq - 1) < ||.seq.(n+1).||/||.seq.n.|| - lim Rseq by SEQ_2:1;
    then 1 - lim Rseq + lim Rseq < ||.seq.(n+1).||/||.seq.n.|| - lim Rseq +
    lim Rseq by XREAL_1:6;
    hence ||.seq.(n+1).||/||.seq.n.|| >= 1;
  end;
  hence thesis by A1,Th30;
end;
