Using results

(i)	R5459: Sample mean of identically distributed random real numbers is an unbiased estimator for expectation
(ii)	R2355: Variance is homogeneous to degree two
(iii)	R2259: Variance of a finite sum of random real numbers

we have \begin{equation} \begin{split} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N (X_n - \overline{X}_N)^2 \right) \\ & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X^2_n - 2 X_n \overline{X}_N + \overline{X}_N^2 \right) \\ & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X^2_n - 2 \overline{X}_N \frac{1}{N} \sum_{n = 1}^N + \overline{X}_N^2 \right) \\ & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X^2_n - 2 \overline{X}_N^2 + \overline{X}_N^2 \right) \\ & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X^2_n - \overline{X}_N^2 \right) \\ & = \mathbb{E} X^2_1 + \mathbb{E} \overline{X}_N^2 \\ & = \sigma^2 + (\mathbb{E} X_1)^2 - \text{Var} \overline{X}_N - (\mathbb{E} \overline{X}_N)^2 \\ & = \sigma^2 + (\mathbb{E} X_1)^2 - \frac{1}{N^2} (N \sigma^2) - (\mathbb{E} X_1)^2 \\ & = \sigma^2 - \frac{1}{N} \sigma^2 \\ & = \frac{N - 1}{N} \sigma^2 \end{split} \end{equation} $\square$