let X be RealUnitarySpace; :: thesis: for seq being sequence of X st seq is convergent holds
||.seq.|| is convergent
let seq be sequence of X; :: thesis: ( seq is convergent implies ||.seq.|| is convergent )
assume seq is convergent ; :: thesis: ||.seq.|| is convergent
then consider g being Point of X such that
A1: for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
||.((seq . n) - g).|| < r by Th9;

now

let r be real number ; :: thesis: ( r > 0 implies ex k being Element of NAT st
for n being Element of NAT st k <= n holds
abs ((||.seq.|| . n) - ||.g.||) < r )
assume A2: r > 0 ; :: thesis: ex k being Element of NAT st
for n being Element of NAT st k <= n holds
abs ((||.seq.|| . n) - ||.g.||) < r
r is Real by XREAL_0:def 1;
then consider m1 being Element of NAT such that
A3: for n being Element of NAT st n >= m1 holds
||.((seq . n) - g).|| < r by A1, A2;
take k = m1; :: thesis: for n being Element of NAT st k <= n holds
abs ((||.seq.|| . n) - ||.g.||) < r

now

let n be Element of NAT ; :: thesis: ( n >= k implies abs ((||.seq.|| . n) - ||.g.||) < r )
assume n >= k ; :: thesis: abs ((||.seq.|| . n) - ||.g.||) < r
then A4: ||.((seq . n) - g).|| < r by A3;
abs (||.(seq . n).|| - ||.g.||) <= ||.((seq . n) - g).|| by BHSP_1:33;
then abs (||.(seq . n).|| - ||.g.||) < r by A4, XXREAL_0:2;
hence abs ((||.seq.|| . n) - ||.g.||) < r by Def3; :: thesis: verum

end;

hence for n being Element of NAT st k <= n holds
abs ((||.seq.|| . n) - ||.g.||) < r ; :: thesis: verum

end;

hence ||.seq.|| is convergent by SEQ_2:def 6; :: thesis: verum