let Rseq be Real_Sequence; :: thesis: for X being RealUnitarySpace
for seq being sequence of X st ( for n being Nat holds
( seq . n <> 0. X & Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| ) ) & Rseq is convergent & lim Rseq < 1 holds
seq is absolutely_summable
let X be RealUnitarySpace; :: thesis: for seq being sequence of X st ( for n being Nat holds
( seq . n <> 0. X & Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| ) ) & Rseq is convergent & lim Rseq < 1 holds
seq is absolutely_summable
let seq be sequence of X; :: thesis: ( ( for n being Nat holds
( seq . n <> 0. X & Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| ) ) & Rseq is convergent & lim Rseq < 1 implies seq is absolutely_summable )
assume that
A1: for n being Nat holds
( seq . n <> H₁(X) & Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| ) and
A2: ( Rseq is convergent & lim Rseq < 1 ) ; :: thesis: seq is absolutely_summable
for n being Nat holds
( ||.seq.|| . n > 0 & Rseq . n = (||.seq.|| . (n + 1)) / (||.seq.|| . n) )

proof

let n be Nat; :: thesis: ( ||.seq.|| . n > 0 & Rseq . n = (||.seq.|| . (n + 1)) / (||.seq.|| . n) )
A3: ||.(seq . n).|| <> 0 by A1, BHSP_1:26;
||.(seq . n).|| >= 0 by BHSP_1:28;
hence ||.seq.|| . n > 0 by A3, BHSP_2:def 3; :: thesis: Rseq . n = (||.seq.|| . (n + 1)) / (||.seq.|| . n)
Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| by A1
.= (||.seq.|| . (n + 1)) / ||.(seq . n).|| by BHSP_2:def 3 ;
hence Rseq . n = (||.seq.|| . (n + 1)) / (||.seq.|| . n) by BHSP_2:def 3; :: thesis: verum

end;

hence seq is absolutely_summable by A2, SERIES_1:26; :: thesis: verum