let RNS be RealNormSpace; :: thesis: for g being Point of RNS
for S being sequence of RNS st S is convergent & lim S = g holds
( ||.(S - g).|| is convergent & lim ||.(S - g).|| = 0 )
let g be Point of RNS; :: thesis: for S being sequence of RNS st S is convergent & lim S = g holds
( ||.(S - g).|| is convergent & lim ||.(S - g).|| = 0 )
let S be sequence of RNS; :: thesis: ( S is convergent & lim S = g implies ( ||.(S - g).|| is convergent & lim ||.(S - g).|| = 0 ) )
assume that
A1: S is convergent and
A2: lim S = g ; :: thesis: ( ||.(S - g).|| is convergent & lim ||.(S - g).|| = 0 )

A3: now

let r be real number ; :: thesis: ( 0 < r implies ex k being Element of NAT st
for n being Element of NAT st k <= n holds
abs ((||.(S - g).|| . n) - 0) < r )
assume A4: 0 < r ; :: thesis: ex k being Element of NAT st
for n being Element of NAT st k <= n holds
abs ((||.(S - g).|| . n) - 0) < r
r is Real by XREAL_0:def 1;
then consider m1 being Element of NAT such that
A5: for n being Element of NAT st m1 <= n holds
||.((S . n) - g).|| < r by A1, A2, A4, Def11;
take k = m1; :: thesis: for n being Element of NAT st k <= n holds
abs ((||.(S - g).|| . n) - 0) < r
let n be Element of NAT ; :: thesis: ( k <= n implies abs ((||.(S - g).|| . n) - 0) < r )
assume k <= n ; :: thesis: abs ((||.(S - g).|| . n) - 0) < r
then ||.((S . n) - g).|| < r by A5;
then A6: ||.(((S . n) - g) - H₁(RNS)).|| < r by RLVECT_1:13;
abs (||.((S . n) - g).|| - ||.H₁(RNS).||) <= ||.(((S . n) - g) - H₁(RNS)).|| by Th13;
then abs (||.((S . n) - g).|| - ||.H₁(RNS).||) < r by A6, XXREAL_0:2;
then abs (||.((S - g) . n).|| - 0) < r by Def7;
hence abs ((||.(S - g).|| . n) - 0) < r by NORMSP_0:def 4; :: thesis: verum

end;

||.(S - g).|| is convergent by A1, Th36, Th39;
hence ( ||.(S - g).|| is convergent & lim ||.(S - g).|| = 0 ) by A3, SEQ_2:def 7; :: thesis: verum