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

proof

defpred S₁[ Nat] means rseq . $1 = (Partial_Sums rseq) . $1;
let m be Nat; :: thesis: (Partial_Sums rseq) . m = 0
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: 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 A4, SERIES_1:def 1 ; :: thesis: verum

end;

A5: S₁[ 0 ] by SERIES_1:def 1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A3);
hence (Partial_Sums rseq) . m = rseq . m
.= 0 by A1 ;
:: thesis: verum

end;

A6: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.(((Partial_Sums rseq) . 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 rseq) . m) - 0).| < p )
assume A7: 0 < p ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.(((Partial_Sums rseq) . m) - 0).| < p
take 0 ; :: thesis: for m being Nat st 0 <= m holds
|.(((Partial_Sums rseq) . m) - 0).| < p
let m be Nat; :: thesis: ( 0 <= m implies |.(((Partial_Sums rseq) . m) - 0).| < p )
assume 0 <= m ; :: thesis: |.(((Partial_Sums rseq) . m) - 0).| < p
|.(((Partial_Sums rseq) . m) - 0).| = |.(0 - 0).| by A2
.= 0 by ABSVALUE:def 1 ;
hence |.(((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 ( rseq is summable & Sum rseq = 0 ) by A8, SERIES_1:def 2, SERIES_1:def 3; :: thesis: verum