let cseq be Complex_Sequence; :: thesis: ( ( for n being Element of NAT holds cseq . n = 0c ) implies ( cseq is absolutely_summable & Sum |.cseq.| = 0 ) )
assume A1: for n being Element of NAT holds cseq . n = 0c ; :: thesis: ( cseq is absolutely_summable & Sum |.cseq.| = 0 )
A2: for n being Element of NAT holds |.cseq.| . n = 0

proof

let n be Element of NAT ; :: thesis: |.cseq.| . n = 0
cseq . n = 0c by A1;
hence |.cseq.| . n = 0 by COMPLEX1:44, VALUED_1:18; :: thesis: verum

end;

A3: for m being Element of NAT holds (Partial_Sums |.cseq.|) . m = 0

proof

defpred S₁[ Element of NAT ] means |.cseq.| . $1 = (Partial_Sums |.cseq.|) . $1;
let m be Element of NAT ; :: thesis: (Partial_Sums |.cseq.|) . m = 0
A4: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: |.cseq.| . k = (Partial_Sums |.cseq.|) . k ; :: thesis: S₁[k + 1]
thus |.cseq.| . (k + 1) = 0 + (|.cseq.| . (k + 1))
.= (|.cseq.| . k) + (|.cseq.| . (k + 1)) by A2
.= (Partial_Sums |.cseq.|) . (k + 1) by A5, SERIES_1:def 1 ; :: thesis: verum

end;

A6: S₁[ 0 ] by SERIES_1:def 1;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A6, A4);
hence (Partial_Sums |.cseq.|) . m = |.cseq.| . m
.= 0 by A2 ;
:: thesis: verum

end;

A7: for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums |.cseq.|) . m) - 0) < p

proof

let p be real number ; :: thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums |.cseq.|) . m) - 0) < p )
assume A8: 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums |.cseq.|) . m) - 0) < p
take 0 ; :: thesis: for m being Element of NAT st 0 <= m holds
abs (((Partial_Sums |.cseq.|) . m) - 0) < p
let m be Element of NAT ; :: thesis: ( 0 <= m implies abs (((Partial_Sums |.cseq.|) . m) - 0) < p )
assume 0 <= m ; :: thesis: abs (((Partial_Sums |.cseq.|) . m) - 0) < p
abs (((Partial_Sums |.cseq.|) . m) - 0) = abs (0 - 0) by A3
.= 0 by ABSVALUE:def 1 ;
hence abs (((Partial_Sums |.cseq.|) . m) - 0) < p by A8; :: thesis: verum

end;

then A9: Partial_Sums |.cseq.| is convergent by SEQ_2:def 6;
then A10: |.cseq.| is summable by SERIES_1:def 2;
lim (Partial_Sums |.cseq.|) = 0 by A7, A9, SEQ_2:def 7;
hence ( cseq is absolutely_summable & Sum |.cseq.| = 0 ) by A10, COMSEQ_3:def 9, SERIES_1:def 3; :: thesis: verum