let X be RealNormSpace; :: thesis: for seq being sequence of X
for rseq1 being Real_Sequence st ( for n being Element of NAT holds rseq1 . n = n -root (||.seq.|| . n) ) & ex m being Element of NAT st
for n being Element of NAT st m <= n holds
rseq1 . n >= 1 holds
not ||.seq.|| is summable
let seq be sequence of X; :: thesis: for rseq1 being Real_Sequence st ( for n being Element of NAT holds rseq1 . n = n -root (||.seq.|| . n) ) & ex m being Element of NAT st
for n being Element of NAT st m <= n holds
rseq1 . n >= 1 holds
not ||.seq.|| is summable
let rseq1 be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds rseq1 . n = n -root (||.seq.|| . n) ) & ex m being Element of NAT st
for n being Element of NAT st m <= n holds
rseq1 . n >= 1 implies not ||.seq.|| is summable )
assume A1: ( ( for n being Element of NAT holds rseq1 . n = n -root (||.seq.|| . n) ) & ex m being Element of NAT st
for n being Element of NAT st m <= n holds
rseq1 . n >= 1 ) ; :: thesis: not ||.seq.|| is summable
for n being Element of NAT holds ||.seq.|| . n >= 0 by Th2;
hence not ||.seq.|| is summable by A1, SERIES_1:29; :: thesis: verum