Anti-Aliasing and the Fourier Transform, Gonzalez Digital Image Processing
Note that in order to obtain the Fourier transform of the windowed time domain signal $f(t)h(t)$, you need to convolve the Fourier transforms of $f(t)$ and $h(t)$. We know that the Fourier transform of $f(t)$ is zero outside some interval (because $f(t)$ is band-limited). However, since $h(t)$ is time-limited, i.e., it is zero outside some time interval, we know that its Fourier transform cannot be band-limited. For this specific example of a rectangular window $h(t)$, we know that the corresponding Fourier transform is a sinc function. Convolving any function with a sinc function results in a function that extends from $-\infty$ to $+\infty$. Consequently, the Fourier transform of the windowed signal $f(t)h(t)$ also extends from $-\infty$ to $+\infty$. Consequently, the signal $f(t)h(t)$ cannot be band-limited.

In sum, a band-limited signal cannot be time-limited, and a time-limited signal cannot be band-limited. Note, however, that you cannot conclude that a signal is band-limited if it is not time-limited and vice versa. There are signal that are neither time-limited nor band-limited.