Demonstrate the mean of the sample variance
The key to prove
$$
E\left[\widehat{\sigma_x^2}\right] = \sigma_x^2 \left(1 - \dfrac{1}{N} \right)
$$
is to add and subtract $\mu_x$ in the expression for the sample variance before working out the square. A detailed proof is available on the Wikipedia page for Bessel's correction.