let X be Banach_Algebra; :: thesis: for seq being sequence of X
for rseq being Real_Sequence st ( for n being Nat holds ||.(seq . n).|| <= rseq . n ) & rseq is convergent & lim rseq = 0 holds
( seq is convergent & lim seq = 0. X )
let seq be sequence of X; :: thesis: for rseq being Real_Sequence st ( for n being Nat holds ||.(seq . n).|| <= rseq . n ) & rseq is convergent & lim rseq = 0 holds
( seq is convergent & lim seq = 0. X )
let rseq be Real_Sequence; :: thesis: ( ( for n being Nat holds ||.(seq . n).|| <= rseq . n ) & rseq is convergent & lim rseq = 0 implies ( seq is convergent & lim seq = 0. X ) )
assume that
A1: for n being Nat holds ||.(seq . n).|| <= rseq . n and
A2: rseq is convergent and
A3: lim rseq = 0 ; :: thesis: ( seq is convergent & lim seq = 0. X )

now :: thesis: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
||.((seq . m) - (0. X)).|| < p

let p be Real; :: thesis: ( 0 < p implies ex n being Nat st
for m being Nat st n <= m holds
||.((seq . m) - (0. X)).|| < p )
assume 0 < p ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
||.((seq . m) - (0. X)).|| < p
then consider n being Nat such that
A4: for m being Nat st n <= m holds
|.((rseq . m) - 0).| < p by A2, A3, SEQ_2:def 7;
reconsider n = n as Nat ;
take n = n; :: thesis: for m being Nat st n <= m holds
||.((seq . m) - (0. X)).|| < p
let m be Nat; :: thesis: ( n <= m implies ||.((seq . m) - (0. X)).|| < p )
assume n <= m ; :: thesis: ||.((seq . m) - (0. X)).|| < p
then A5: |.((rseq . m) - 0).| < p by A4;
A6: ||.((seq . m) - (0. X)).|| = ||.(seq . m).|| by RLVECT_1:13;
A7: rseq . m <= |.(rseq . m).| by ABSVALUE:4;
||.(seq . m).|| <= rseq . m by A1;
then ||.((seq . m) - (0. X)).|| <= |.(rseq . m).| by A6, A7, XXREAL_0:2;
hence ||.((seq . m) - (0. X)).|| < p by A5, XXREAL_0:2; :: thesis: verum

end;

then A8: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
||.((seq . m) - (0. X)).|| < p ;
hence seq is convergent ; :: thesis: lim seq = 0. X
hence lim seq = 0. X by A8, NORMSP_1:def 7; :: thesis: verum