
theorem
  for X be ComplexNormSpace, seq be sequence of X, rseq1 be
Real_Sequence holds (for n be Nat holds seq.n<>0.X & rseq1.n=||.seq
  .||.(n+1)/||.seq.||.n) & rseq1 is convergent & lim rseq1 < 1 implies seq is
  norm_summable
proof
  let X be ComplexNormSpace;
  let seq be sequence of X;
  let rseq1 be Real_Sequence;
  assume that
A1: for n be Nat holds seq.n<>0.X & rseq1.n=||.seq.||.(n+1)/
  ||. seq.||.n and
A2: rseq1 is convergent & lim rseq1 < 1;
  now
    let n be Nat;
A3: 0 <=||.seq.||.n by Th2;
A4: 0 <=||.seq.||.(n+1) by Th2;
A5: abs((||.seq.||)).(n+1) =|.(||.seq.||).(n+1).| by SEQ_1:12
      .=||.seq.||.(n+1) by A4,ABSVALUE:def 1;
A6: seq.n<>0.X & ||.seq.||.n =||.seq.n.|| by A1,NORMSP_0:def 4;
    abs((||.seq.||)).n =|.(||.seq.||).n.| by SEQ_1:12
      .=||.seq.||.n by A3,ABSVALUE:def 1;
    hence (||.seq.||).n <>0 & rseq1.n=abs((||.seq.||)).(n+1)/abs((||.seq.||)).
    n by A1,A5,A6,NORMSP_0:def 5;
  end;
  then ||.seq.|| is absolutely_summable by A2,SERIES_1:37;
  then
A7: abs(||.seq.||) is summable;
  now
    let n be Element of NAT;
A8: 0 <=||.seq.||.n by Th2;
    thus abs((||.seq.||)).n =|.(||.seq.||).n.| by SEQ_1:12
      .=||.seq.||.n by A8,ABSVALUE:def 1;
  end;
  then abs((||.seq.||)) =||.seq.|| by FUNCT_2:63;
  hence thesis by A7;
end;
