This is a classic theorem in game theory, that is left as an excersice in my textbook. Can anybody proove it? I can not thing of anything excpet from the definition of the correlated equilibrium in first place. Here is the theorem and the definition as well.
$\mathbf{Theorem:}$ For every Nash equilibrium $\sigma^*$, the probability distribution $p_{\sigma^*}$ is a correlated equilibrium.
$\mathbf{Definition:}$ A probability distribution $p$ over the set of action vectors $S$ is called a correlated equilibrium if the strategy vector $\tau^*$ is a Nash equilibrium of the game $\Gamma^*(p)$. In other words, for every player $i ∈ N$:
\begin{equation}\Sigma_{s_{-i}\in S_{-i}}p(s_i,s_{-i})u_i(s_i,s_{-i})\geq \Sigma_{s_{-i}\in S_{-i}}p(s_i,s_{-i})u_i(s^{'}_i,s_{-i}),\quad\text{$\forall s_i,s^{'}_i\in S_i$}\end{equation}
Every strategy vector $\sigma$ induces a probability distribution $p_{\sigma^*}$ over the set of action vector $S$. \begin{equation}p_{\sigma^*}(s_1,...,s_n)=\sigma_1(s_1)\times\sigma_2(s_2)\times...\times\sigma_n(s_n)\end{equation}
$\textit{Hint:}$ When we relate to a Nash equilibrium $\sigma^*$ as a correlated equilibrium we mean the probability distribution $p_{\sigma^*}$ given by the aforementioned equation: