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

ex n being Nat st

for m being Nat st n <= m holds

|.(((Partial_Sums rseq) . m) - 0).| < p

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

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

A6:
for p being Real st 0 < p holds
defpred S_{1}[ Nat] means rseq . $1 = (Partial_Sums rseq) . $1;

let m be Nat; :: thesis: (Partial_Sums rseq) . m = 0

A3: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]
_{1}[ 0 ]
by SERIES_1:def 1;

for n being Nat holds S_{1}[n]
from NAT_1:sch 2(A5, A3);

hence (Partial_Sums rseq) . m = rseq . m

.= 0 by A1 ;

:: thesis: verum

end;let m be Nat; :: thesis: (Partial_Sums rseq) . m = 0

A3: for k being Nat st S

S

proof

A5:
S
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A4: rseq . k = (Partial_Sums rseq) . k ; :: thesis: S_{1}[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;assume A4: rseq . k = (Partial_Sums rseq) . k ; :: thesis: S

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

for n being Nat holds S

hence (Partial_Sums rseq) . m = rseq . m

.= 0 by A1 ;

:: thesis: verum

ex n being Nat st

for m being Nat st n <= m holds

|.(((Partial_Sums rseq) . m) - 0).| < p

proof

then A8:
Partial_Sums rseq is convergent
by SEQ_2:def 6;
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;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

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