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

proof

let n be 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 Nat holds (Partial_Sums |.cseq.|) . m = 0

proof

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

proof

let k be 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 Nat holds S₁[n] from NAT_1:sch 2(A6, A4);
hence (Partial_Sums |.cseq.|) . m = |.cseq.| . m
.= 0 by A2 ;
:: thesis: verum

end;

A7: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.(((Partial_Sums |.cseq.|) . m) - 0).| < p

proof

let p be Real; :: thesis: ( 0 < p implies ex n being Nat st
for m being Nat st n <= m holds
|.(((Partial_Sums |.cseq.|) . m) - 0).| < p )
assume A8: 0 < p ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.(((Partial_Sums |.cseq.|) . m) - 0).| < p
take 0 ; :: thesis: for m being Nat st 0 <= m holds
|.(((Partial_Sums |.cseq.|) . m) - 0).| < p
let m be Nat; :: thesis: ( 0 <= m implies |.(((Partial_Sums |.cseq.|) . m) - 0).| < p )
assume 0 <= m ; :: thesis: |.(((Partial_Sums |.cseq.|) . m) - 0).| < p
|.(((Partial_Sums |.cseq.|) . m) - 0).| = |.(0 - 0).| by A3
.= 0 by ABSVALUE:def 1 ;
hence |.(((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