Exercise 3.3 Discussion

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf {Exercise\ 3.3}$
Show that the state

(1)
\begin{align} \ket{W_n} = \frac{1}{\sqrt{n}} (\ket{0\dots 001} + \ket{0\dots 010} + \ket{0\dots 100} + \cdots + \ket{1\dots 000}) \end{align}

is entangled, with respect to the decompostion into the $n$ qubits, for every $n>1$.