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

proof

let m be Element of NAT ; :: thesis: (Partial_Sums rseq) . m = 0
defpred S₁[ Nat] means rseq . $1 = (Partial_Sums rseq) . $1;
A3: S₁[ 0 ] by SERIES_1:def 1;
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: rseq . k = (Partial_Sums rseq) . k ; :: thesis: S₁[k + 1]
thus rseq . (k + 1) = 0 + (rseq . (k + 1))
.= (rseq . k) + (rseq . (k + 1)) by A1
.= (Partial_Sums rseq) . (k + 1) by A5, SERIES_1:def 1 ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A3, A4);
hence (Partial_Sums rseq) . m = rseq . m
.= 0 by A1 ;
:: thesis: verum

end;

( Sum rseq = 0 & rseq is summable )

proof

A6: 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 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 rseq) . m) - 0 ) < p )
assume A7: 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums rseq) . m) - 0 ) < p
take 0 ; :: thesis: for m being Element of NAT st 0 <= m holds
abs (((Partial_Sums rseq) . m) - 0 ) < p
let m be Element of NAT ; :: thesis: ( 0 <= m implies abs (((Partial_Sums rseq) . m) - 0 ) < p )
assume 0 <= m ; :: thesis: abs (((Partial_Sums rseq) . m) - 0 ) < p
abs (((Partial_Sums rseq) . m) - 0 ) = abs (0 - 0 ) by A2
.= 0 by ABSVALUE:def 1 ;
hence abs (((Partial_Sums rseq) . m) - 0 ) < p by A7; :: thesis: verum

end;

then A8: Partial_Sums rseq is convergent by SEQ_2:def 6;
then lim (Partial_Sums rseq) = 0 by A6, SEQ_2:def 7;
hence ( Sum rseq = 0 & rseq is summable ) by A8, SERIES_1:def 2, SERIES_1:def 3; :: thesis: verum

end;

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