let rseq be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds rseq . n = 0 ) implies ( rseq is absolutely_summable & Sum (abs rseq) = 0 ) )
assume A1: for n being Element of NAT holds rseq . n = 0 ; :: thesis: ( rseq is absolutely_summable & Sum (abs rseq) = 0 )
A2: for n being Element of NAT holds (abs rseq) . n = 0

proof

let n be Element of NAT ; :: thesis: (abs rseq) . n = 0
A3: rseq . n = 0 by A1;
(abs rseq) . n = abs (rseq . n) by SEQ_1:16;
hence (abs rseq) . n = 0 by A3, ABSVALUE:7; :: thesis: verum

end;

A4: for m being Element of NAT holds (Partial_Sums (abs rseq)) . m = 0

proof

let m be Element of NAT ; :: thesis: (Partial_Sums (abs rseq)) . m = 0
defpred S₁[ Element of NAT ] means (abs rseq) . $1 = (Partial_Sums (abs rseq)) . $1;
A5: S₁[ 0 ] by SERIES_1:def 1;
A6: 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 A7: (abs rseq) . k = (Partial_Sums (abs rseq)) . k ; :: thesis: S₁[k + 1]
thus (abs rseq) . (k + 1) = 0 + ((abs rseq) . (k + 1))
.= ((abs rseq) . k) + ((abs rseq) . (k + 1)) by A2
.= (Partial_Sums (abs rseq)) . (k + 1) by A7, SERIES_1:def 1 ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A5, A6);
hence (Partial_Sums (abs rseq)) . m = (abs rseq) . m
.= 0 by A2 ;
:: thesis: verum

end;

( Sum (abs rseq) = 0 & rseq is absolutely_summable )

proof

A8: 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 (abs rseq)) . 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 (abs rseq)) . m) - 0 ) < p )
assume A9: 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums (abs rseq)) . m) - 0 ) < p
take 0 ; :: thesis: for m being Element of NAT st 0 <= m holds
abs (((Partial_Sums (abs rseq)) . m) - 0 ) < p
let m be Element of NAT ; :: thesis: ( 0 <= m implies abs (((Partial_Sums (abs rseq)) . m) - 0 ) < p )
assume 0 <= m ; :: thesis: abs (((Partial_Sums (abs rseq)) . m) - 0 ) < p
abs (((Partial_Sums (abs rseq)) . m) - 0 ) = abs (0 - 0 ) by A4
.= 0 by ABSVALUE:def 1 ;
hence abs (((Partial_Sums (abs rseq)) . m) - 0 ) < p by A9; :: thesis: verum

end;

then A10: Partial_Sums (abs rseq) is convergent by SEQ_2:def 6;
then A11: abs rseq is summable by SERIES_1:def 2;
lim (Partial_Sums (abs rseq)) = 0 by A8, A10, SEQ_2:def 7;
hence ( Sum (abs rseq) = 0 & rseq is absolutely_summable ) by A11, SERIES_1:def 3, SERIES_1:def 5; :: thesis: verum

end;

hence ( rseq is absolutely_summable & Sum (abs rseq) = 0 ) ; :: thesis: verum