Product of power signal and energy signal
If you are not looking for a rigorous proof, then the following would suffice for a particular class of power signals which has the property that their magnitude is bounded; i.e, $ |x_p(t)| \leq K$ for all $t$.

Numerous examples exist for such power signals like sinusoidal waves, complex exponentials, all kinds of periodic continuous-time signals which are defined via convergent Fourier series sums... Then for such a power signal $x_p(t)$ and a given energy signal $x_e(t)$ you can show that their product $y(t) = x_p(t) x_e(t)$ will be an energy signal as follows; given

$$ E_x = \int_{-\infty}^{\infty} |x_e(t)|^2 dt $$

and $ |x_p(t)| \leq K $ for all $t$ then the energy of y(t) is

$$ E_y = \int_{-\infty}^{\infty} |y(t)|^2 dt = \int_{-\infty}^{\infty} |x_p(t) x_e(t)|^2 dt \leq \int_{-\infty}^{\infty} K^2 |x_e(t)|^2 dt = K^2 E_x$$

Hence $y(t)$ will also be an energy signal.