let CNS be ComplexNormSpace; :: thesis: for S being sequence of CNS st S is convergent holds
||.S.|| is convergent
let S be sequence of CNS; :: thesis: ( S is convergent implies ||.S.|| is convergent )
assume S is convergent ; :: thesis: ||.S.|| is convergent
then consider g being Point of CNS such that
A1: for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - g).|| < r ;

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

let r be Real; :: thesis: ( 0 < r implies ex k being Nat st
for n being Nat st k <= n holds
|.((||.S.|| . n) - ||.g.||).| < r )
assume A2: 0 < r ; :: thesis: ex k being Nat st
for n being Nat st k <= n holds
|.((||.S.|| . n) - ||.g.||).| < r
consider m1 being Nat such that
A3: for n being Nat st m1 <= n holds
||.((S . n) - g).|| < r by A1, A2;
take k = m1; :: thesis: for n being Nat st k <= n holds
|.((||.S.|| . n) - ||.g.||).| < r
let n be Nat; :: thesis: ( k <= n implies |.((||.S.|| . n) - ||.g.||).| < r )
assume k <= n ; :: thesis: |.((||.S.|| . n) - ||.g.||).| < r
then A4: ||.((S . n) - g).|| < r by A3;
|.(||.(S . n).|| - ||.g.||).| <= ||.((S . n) - g).|| by Th110;
then |.(||.(S . n).|| - ||.g.||).| < r by A4, XXREAL_0:2;
hence |.((||.S.|| . n) - ||.g.||).| < r by NORMSP_0:def 4; :: thesis: verum

end;

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