let X be RealNormSpace; :: thesis: for rseq1 being Real_Sequence
for seq2 being sequence of X st rseq1 is summable & ex m being Element of NAT st
for n being Element of NAT st m <= n holds
||.(seq2 . n).|| <= rseq1 . n holds
seq2 is norm_summable
let rseq1 be Real_Sequence; :: thesis: for seq2 being sequence of X st rseq1 is summable & ex m being Element of NAT st
for n being Element of NAT st m <= n holds
||.(seq2 . n).|| <= rseq1 . n holds
seq2 is norm_summable
let seq2 be sequence of X; :: thesis: ( rseq1 is summable & ex m being Element of NAT st
for n being Element of NAT st m <= n holds
||.(seq2 . n).|| <= rseq1 . n implies seq2 is norm_summable )
assume A1: ( rseq1 is summable & ex m being Element of NAT st
for n being Element of NAT st m <= n holds
||.(seq2 . n).|| <= rseq1 . n ) ; :: thesis: seq2 is norm_summable
then consider m being Element of NAT such that
A2: for n being Element of NAT st m <= n holds
||.(seq2 . n).|| <= rseq1 . n ;

A3: now

let n be Element of NAT ; :: thesis: ( m <= n implies ||.seq2.|| . n <= rseq1 . n )
assume m <= n ; :: thesis: ||.seq2.|| . n <= rseq1 . n
then ||.(seq2 . n).|| <= rseq1 . n by A2;
hence ||.seq2.|| . n <= rseq1 . n by NORMSP_1:def 10; :: thesis: verum

end;

now

let n be Element of NAT ; :: thesis: 0 <= ||.seq2.|| . n
||.(seq2 . n).|| = ||.seq2.|| . n by NORMSP_1:def 10;
hence 0 <= ||.seq2.|| . n by NORMSP_1:8; :: thesis: verum

end;

hence ||.seq2.|| is summable by A1, A3, SERIES_1:22; :: according to LOPBAN_3:def 4 :: thesis: verum