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

A3: now :: thesis: for r being Real st r > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
|.((||.(seq - g).|| . n) - 0).| < r

let r be Real; :: thesis: ( r > 0 implies ex k being Nat st
for n being Nat st n >= k holds
|.((||.(seq - g).|| . n) - 0).| < r )
assume A4: r > 0 ; :: thesis: ex k being Nat st
for n being Nat st n >= k holds
|.((||.(seq - g).|| . n) - 0).| < r
consider m1 being Nat such that
A5: for n being Nat st n >= m1 holds
||.((seq . n) - g).|| < r by A1, A2, A4, Th19;
reconsider k = m1 as Nat ;
take k = k; :: thesis: for n being Nat st n >= k holds
|.((||.(seq - g).|| . n) - 0).| < r
let n be Nat; :: thesis: ( n >= k implies |.((||.(seq - g).|| . n) - 0).| < r )
assume n >= k ; :: thesis: |.((||.(seq - g).|| . n) - 0).| < r
then ||.((seq . n) - g).|| < r by A5;
then A6: ||.(((seq . n) - g) - H₁(X)).|| < r by RLVECT_1:13;
|.(||.((seq . n) - g).|| - ||.H₁(X).||).| <= ||.(((seq . n) - g) - H₁(X)).|| by BHSP_1:33;
then |.(||.((seq . n) - g).|| - ||.H₁(X).||).| < r by A6, XXREAL_0:2;
then |.(||.((seq . n) - g).|| - 0).| < r by BHSP_1:26;
then |.(||.((seq - g) . n).|| - 0).| < r by NORMSP_1:def 4;
hence |.((||.(seq - g).|| . n) - 0).| < r by Def3; :: thesis: verum

end;

||.(seq - g).|| is convergent by A1, Th8, Th20;
hence ( ||.(seq - g).|| is convergent & lim ||.(seq - g).|| = 0 ) by A3, SEQ_2:def 7; :: thesis: verum