let X be RealNormSpace; :: thesis: for seq being sequence of X

for rseq1 being Real_Sequence st ( for n being Nat holds rseq1 . n = n -root (||.seq.|| . n) ) & rseq1 is convergent & lim rseq1 > 1 holds

not seq is norm_summable

let seq be sequence of X; :: thesis: for rseq1 being Real_Sequence st ( for n being Nat holds rseq1 . n = n -root (||.seq.|| . n) ) & rseq1 is convergent & lim rseq1 > 1 holds

not seq is norm_summable

let rseq1 be Real_Sequence; :: thesis: ( ( for n being Nat holds rseq1 . n = n -root (||.seq.|| . n) ) & rseq1 is convergent & lim rseq1 > 1 implies not seq is norm_summable )

assume A1: ( ( for n being Nat holds rseq1 . n = n -root (||.seq.|| . n) ) & rseq1 is convergent & lim rseq1 > 1 ) ; :: thesis: not seq is norm_summable

for n being Nat holds ||.seq.|| . n >= 0 by Th2;

hence not ||.seq.|| is summable by A1, SERIES_1:30; :: according to LOPBAN_3:def 3 :: thesis: verum

for rseq1 being Real_Sequence st ( for n being Nat holds rseq1 . n = n -root (||.seq.|| . n) ) & rseq1 is convergent & lim rseq1 > 1 holds

not seq is norm_summable

let seq be sequence of X; :: thesis: for rseq1 being Real_Sequence st ( for n being Nat holds rseq1 . n = n -root (||.seq.|| . n) ) & rseq1 is convergent & lim rseq1 > 1 holds

not seq is norm_summable

let rseq1 be Real_Sequence; :: thesis: ( ( for n being Nat holds rseq1 . n = n -root (||.seq.|| . n) ) & rseq1 is convergent & lim rseq1 > 1 implies not seq is norm_summable )

assume A1: ( ( for n being Nat holds rseq1 . n = n -root (||.seq.|| . n) ) & rseq1 is convergent & lim rseq1 > 1 ) ; :: thesis: not seq is norm_summable

for n being Nat holds ||.seq.|| . n >= 0 by Th2;

hence not ||.seq.|| is summable by A1, SERIES_1:30; :: according to LOPBAN_3:def 3 :: thesis: verum