Why is this a Parseval Frame? In Introduction to Finite Frame Theory it is given without proof that the system
$\left \{\frac{e_1}{1}, \frac{e_2}{\sqrt{2}}, \frac{e_2}{\sqrt{2}}, \frac{e_3}{\sqrt{3}}, \frac{e_3}{\sqrt{3}}, \frac{e_3}{\sqrt{3}}, ..., \frac{e_N}{\sqrt{N}}, \frac{e_N}{\sqrt{N}}, ..., \frac{e_N}{\sqrt{N}}\right \}$
is a Parseval frame for $\mathcal{H}^N = \mathbb{C}^N$ with $\{e_n\}_{n = 1}^N$ being the ONB.
Why is that? Maybe it's just too simple and I cannot see it ...

Do you know Parseval's identity? It states that for an ONB $\{e_n\}_{n=1}^N$, we have that
\begin{equation} \|x\|^2=\sum_{n=1}^N|\langle x,e_n\rangle|^2,\quad \forall x\in \mathbb{C}^N.\end{equation}
The claim follows almost immediately from this identity since it holds for any $x\in \mathbb{C}^N$ that
\begin{align}
|\langle x, e_1\rangle|^2+|\langle x, \frac{1}{\sqrt{2}}e_2\rangle|^2+|\langle x, \frac{1}{\sqrt{2}}e_2\rangle|^2+\cdots&=\sum_{n=1}^N n |\langle x, \frac{1}{\sqrt{n}}e_n\rangle|^2\\
&=\sum_{n=1}^N n\cdot \frac{1}{n} |\langle x,e_n\rangle|^2\\
&=\sum_{n=1}^N |\langle x,e_n\rangle|^2\\
&=\|x\|^2.
\end{align}
