If the expected value and variance of an estimator are known, then it is rather easy to prove convergence by combining the following two theorems.
Theorem: If $\x_n \overset{r}{\longrightarrow} \x$ for some $r > 0$, then $\x_n \inP \x$.
Proof: Homework assignment.
Theorem: If $\a \in \real^d$, then $\x_n \inQM \a$ if and only if $\Ex \x_n \to \a$ and $\Var \x_n \to \zero$.
Proof: Homework assignment.
In other words, $\bth$ is a consistent estimator of $\bt$ if $\Ex\bth \to \bt$ and $\Var\bth \to \zero$.