Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3, \ldots$ is an D3842: Uncorrelated random collection
(ii)	\begin{equation} \exists \, \mu \in \mathbb{R} : \forall \, n \in 1, 2, 3, \ldots : \mathbb{E} X_n = \mu \end{equation}
(iii)	\begin{equation} \exists \, C \in [0, \infty) : \forall \, n \in 1, 2, 3, \ldots : \text{Var} X_n \leq C \end{equation}

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. $\square$