A sketch of the proof indeed uses the Cauchy-Schwarz inequality:

$$ \left|\int_\mathbb{R} x(t'-t)h(t)\mathrm{d}t\right| \le \int_\mathbb{R} \left| x(t'-t)\right|\left|h(t)\right|\mathrm{d}t\le \left(\int_\mathbb{R} \left| x(t'-t)\right|^2\mathrm{d}t\right)^{1/2}\left(\int_\mathbb{R}\left|h(t)\right|^2\mathrm{d}t\right)^{1/2}$$
which is finite since $x$ and $h$ are $L_2$ or "energy signals".

This shape of proof extends to functions in $L_p$ and $L_q$ respectively as long as $p$ and $q$ are conjugated:

$$ \frac{1}{p}+ \frac{1}{q}=1\,.$$