Let $X_1$, $X_2$, ... be iid random variables with $E(X_1)$ = $\mu < \infty$ and $Var(X_1)$ = $\sigma^2 < \infty$.
Does the Central Limit Theorem implies that the sample mean converges in distribution to a normally distributed random variable?
Also, does the theorem implies that the sample mean is a consistent estimator for $\mu$?