= Solution
Let $m=[G:H]$, extend $\psi$ to $\psi_0$ by zero outside $H$, and choose <coset> representatives $x_1,\ldots,x_m$. Then $\operatorname{Ind}\psi(g)=\sum_{j=1}^m f_j(g)$ with $f_j(g)=\psi_0(x_j^{-1}gx_j)$. Each summand has squared $G$-norm
$$
\|f_j\|_G^2=\frac1{|G|}\sum_{h\in H}|\psi(h)|^2=\frac1m,
$$
since $\psi$ is irreducible. Pointwise <Cauchy-Schwarz inequality> gives $|\sum f_j|^2\le m\sum|f_j|^2$. After summing and dividing by $|G|$,
$$
\boxed{\sum_i e_i^2=\|\operatorname{Ind}\psi\|_G^2\le m\sum_j\|f_j\|_G^2=m.}
$$
The $e_i$ are nonnegative integers by part (b). This proves the <character induction norm bound> without an assumption that $H$ is normal.
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