Given the following Law of Large numbers: Let $X_t$ be a sequence of random variables with $E[X_t] = \mu, |cov(X_t,X_s)| \leq c * g^{|t-s|}$, where $g \in (0,1)$, then: $$ \frac{1}{T} \sum_{t=1}^{T} X_t \rightarrow \mu $$ in probability.
Moreover, given an AR(1) process: $$ y_t = \alpha + g*y_{t-1} + u_t $$ where |g| < 1.
Show that: $$ \frac{1}{T} \sum_{t=1}^{T} X_t \rightarrow \frac{mu}{1-g} $$
I was thinking about going through showing the convergence to zero of the $V(\frac{1}{T} \sum_{t=1}^{T} (X_t - \mu)$, but i don't know how to proceed. ANy suggestion?