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