let Rseq be Real_Sequence; :: thesis: for X being RealUnitarySpace
for seq being sequence of X st ( for n being Nat holds Rseq . n = n -root ||.(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 Rseq . n = n -root ||.(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 Rseq . n = n -root ||.(seq . n).|| ) & Rseq is convergent & lim Rseq < 1 implies seq is absolutely_summable )
assume that
A1: for n being Nat holds Rseq . n = n -root ||.(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 = n -root (||.seq.|| . n) )
proof
let n be Nat; :: thesis: ( ||.seq.|| . n >= 0 & Rseq . n = n -root (||.seq.|| . n) )
||.seq.|| . n = ||.(seq . n).|| by BHSP_2:def 3;
hence ||.seq.|| . n >= 0 by BHSP_1:28; :: thesis: Rseq . n = n -root (||.seq.|| . n)
Rseq . n = n -root ||.(seq . n).|| by A1;
hence Rseq . n = n -root (||.seq.|| . n) by BHSP_2:def 3; :: thesis: verum
end;
hence seq is absolutely_summable by A2, SERIES_1:28; :: thesis: verum