let G be RealNormSpace; :: thesis: for s1 being Real_Sequence
for seq being sequence of G st ( for n being Element of NAT holds ||.(seq . n).|| <= s1 . n ) & s1 is convergent & lim s1 = 0 holds
( seq is convergent & lim seq = 0. G )
let s1 be Real_Sequence; :: thesis: for seq being sequence of G st ( for n being Element of NAT holds ||.(seq . n).|| <= s1 . n ) & s1 is convergent & lim s1 = 0 holds
( seq is convergent & lim seq = 0. G )
let seq be sequence of G; :: thesis: ( ( for n being Element of NAT holds ||.(seq . n).|| <= s1 . n ) & s1 is convergent & lim s1 = 0 implies ( seq is convergent & lim seq = 0. G ) )
assume that
A1: for n being Element of NAT holds ||.(seq . n).|| <= s1 . n and
A2: s1 is convergent and
A3: lim s1 = 0 ; :: thesis: ( seq is convergent & lim seq = 0. G )

now

let p be real number ; :: thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((seq . m) - (0. G)).|| < p )
assume 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((seq . m) - (0. G)).|| < p
then consider n being Element of NAT such that
A4: for m being Element of NAT st n <= m holds
abs ((s1 . m) - 0) < p by A2, A3, SEQ_2:def 7;

now

let m be Element of NAT ; :: thesis: ( n <= m implies ||.((seq . m) - (0. G)).|| < p )
assume n <= m ; :: thesis: ||.((seq . m) - (0. G)).|| < p
then A5: abs ((s1 . m) - 0) < p by A4;
A6: ||.((seq . m) - (0. G)).|| = ||.(seq . m).|| by RLVECT_1:13;
A7: s1 . m <= abs (s1 . m) by ABSVALUE:4;
||.(seq . m).|| <= s1 . m by A1;
then ||.((seq . m) - (0. G)).|| <= abs (s1 . m) by A6, A7, XXREAL_0:2;
hence ||.((seq . m) - (0. G)).|| < p by A5, XXREAL_0:2; :: thesis: verum

end;

hence ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((seq . m) - (0. G)).|| < p ; :: thesis: verum

end;

then A8: for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((seq . m) - (0. G)).|| < p ;
hence seq is convergent by NORMSP_1:def 6; :: thesis: lim seq = 0. G
hence lim seq = 0. G by A8, NORMSP_1:def 7; :: thesis: verum