let p be Real; :: thesis: ( p >= 1 implies for rseq being Real_Sequence st ( for n being Element of NAT holds rseq . n = 0 ) holds
( rseq rto_power p is summable & (Sum (rseq rto_power p)) to_power (1 / p) = 0 ) )
assume A1: p >= 1 ; :: thesis: for rseq being Real_Sequence st ( for n being Element of NAT holds rseq . n = 0 ) holds
( rseq rto_power p is summable & (Sum (rseq rto_power p)) to_power (1 / p) = 0 )
A2: 1 / p > 0 by A1, XREAL_1:141;
let rseq be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds rseq . n = 0 ) implies ( rseq rto_power p is summable & (Sum (rseq rto_power p)) to_power (1 / p) = 0 ) )
assume A3: for n being Element of NAT holds rseq . n = 0 ; :: thesis: ( rseq rto_power p is summable & (Sum (rseq rto_power p)) to_power (1 / p) = 0 )
A4: for n being Element of NAT holds (rseq rto_power p) . n = 0

proof

let n be Element of NAT ; :: thesis: (rseq rto_power p) . n = 0
rseq . n = 0 by A3;
then abs (rseq . n) = 0 by ABSVALUE:7;
then (abs (rseq . n)) to_power p = 0 by A1, POWER:def 2;
hence (rseq rto_power p) . n = 0 by Def1; :: thesis: verum

end;

A5: for m being Element of NAT holds (Partial_Sums (rseq rto_power p)) . m = 0

proof

let m be Element of NAT ; :: thesis: (Partial_Sums (rseq rto_power p)) . m = 0
defpred S₁[ Element of NAT ] means (rseq rto_power p) . $1 = (Partial_Sums (rseq rto_power p)) . $1;
A6: S₁[ 0 ] by SERIES_1:def 1;
A7: 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 A8: (rseq rto_power p) . k = (Partial_Sums (rseq rto_power p)) . k ; :: thesis: S₁[k + 1]
thus (rseq rto_power p) . (k + 1) = 0 + ((rseq rto_power p) . (k + 1))
.= ((rseq rto_power p) . k) + ((rseq rto_power p) . (k + 1)) by A4
.= (Partial_Sums (rseq rto_power p)) . (k + 1) by A8, SERIES_1:def 1 ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A6, A7);
hence (Partial_Sums (rseq rto_power p)) . m = (rseq rto_power p) . m
.= 0 by A4 ;
:: thesis: verum

end;

( Sum (rseq rto_power p) = 0 & rseq rto_power p is summable )

proof

A9: for e being real number st 0 < e holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums (rseq rto_power p)) . m) - 0 ) < e

proof

let e be real number ; :: thesis: ( 0 < e implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums (rseq rto_power p)) . m) - 0 ) < e )
assume A10: 0 < e ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums (rseq rto_power p)) . m) - 0 ) < e
take 0 ; :: thesis: for m being Element of NAT st 0 <= m holds
abs (((Partial_Sums (rseq rto_power p)) . m) - 0 ) < e
let m be Element of NAT ; :: thesis: ( 0 <= m implies abs (((Partial_Sums (rseq rto_power p)) . m) - 0 ) < e )
assume 0 <= m ; :: thesis: abs (((Partial_Sums (rseq rto_power p)) . m) - 0 ) < e
abs (((Partial_Sums (rseq rto_power p)) . m) - 0 ) = abs (0 - 0 ) by A5
.= 0 by ABSVALUE:def 1 ;
hence abs (((Partial_Sums (rseq rto_power p)) . m) - 0 ) < e by A10; :: thesis: verum

end;

then A11: Partial_Sums (rseq rto_power p) is convergent by SEQ_2:def 6;
then lim (Partial_Sums (rseq rto_power p)) = 0 by A9, SEQ_2:def 7;
hence ( Sum (rseq rto_power p) = 0 & rseq rto_power p is summable ) by A11, SERIES_1:def 2, SERIES_1:def 3; :: thesis: verum

end;

hence ( rseq rto_power p is summable & (Sum (rseq rto_power p)) to_power (1 / p) = 0 ) by A2, POWER:def 2; :: thesis: verum