let rseq1 be Real_Sequence; :: thesis: for seq2 being Complex_Sequence 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 absolutely_summable
let seq2 be Complex_Sequence; :: 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 absolutely_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 absolutely_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 VALUED_1:18; :: thesis: verum

end;

now

let n be Element of NAT ; :: thesis: 0 <= |.seq2.| . n
|.(seq2 . n).| = |.seq2.| . n by VALUED_1:18;
hence 0 <= |.seq2.| . n by COMPLEX1:132; :: thesis: verum

end;

hence |.seq2.| is summable by A1, A3, SERIES_1:22; :: according to COMSEQ_3:def 11 :: thesis: verum