Fix a positive integer $N \in 1, 2, 3 \ldots$ and set $S_N : = \sum_{n = 1}^N X_n$. Applying
R4652: Real-linearity of real expectation, we have
\begin{equation}
\mathbb{E}(S_N / N) = \frac{1}{N} \sum_{n = 1}^N \mathbb{E} X_n = \frac{1}{N} \sum_{n = 1}^N \mu = \mu
\end{equation}
Since $X_1, X_2, X_3, \dots$ are uncorrelated, we can use result
R2259: Variance of a finite sum of random real numbers to obtain
\begin{equation}
\mathbb{E}(|S_N / N - \mu|^2) = : \text{Var}(S_N / N) = \frac{1}{N^2} \sum_{n = 1}^N \text{Var}(X_n) \leq \frac{1}{N^2} \sum_{n = 1}^N C = \frac{1}{N} C
\end{equation}
Taking limits on each side as $N \to \infty$ and applying
R1096: Squeeze theorem for basic sequences, we conclude
\begin{equation}
0 \leq \lim_{N \to \infty} \mathbb{E}(|S_N / N - \mu|^2) \leq \lim_{N \to \infty} \frac{1}{N} C = 0
\end{equation}
This settles the first claim. The second claim is now a consequence of
R2390: Convergence in absolute moment implies convergence in probability. $\square$
