Time-Bandwidth Product
Note that these definitions of effective duration and effective bandwidth are only useful for very specific signals, namely for (real-valued) low-pass signals that are even and centered around $t=0$. This implies that their Fourier transform is also real-valued, even and centered around $\omega=0$, and, consequently, the same definition of width can be used. 

Now for the proof, which is straightforward. Take the definition of the Fourier transform and its inverse:

$$F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\tag{1}$$

$$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{j\omega t}d\omega\tag{2}$$

From $(1)$ and $(2)$ we get

$$F(0)=\int_{-\infty}^{\infty}f(t)dt\tag{3}$$

and

$$f(0)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)d\omega\tag{4}$$

from which we get

$$T_e=\frac{F(0)}{f(0)}\tag{5}$$

$$B_e=\frac{f(0)}{F(0)}\tag{6}$$

The result follows immediately.

For information on more general definitions of duration/bandwidth and on the uncertainty principle take a look at this answer.