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

(i)	$X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
(ii)	$$\begin{equation} \lim_{x \to \infty} x \mathbb{P}(|X_1| > x) = 0 \end{equation}$$
(iii)	$$\begin{equation} \mu_N : = \mathbb{E}(X_1 I_{\{ |X_1| \leq N \}}) \end{equation}$$

The idea of the proof is to apply R3260: Weak law of large numbers for random real triangular arrays for a triangular array $$\begin{equation} \begin{split} & X_1 \\ & X_1 \quad X_2 \\ & X_1 \quad X_2 \quad X_3 \\ & \quad \vdots \end{split} \end{equation}$$ where, in each column, the variables are the one and the same variable and we have $\lambda_N : = N$. To this end, we need to confirm that the assumptions of the theorem in this special case hold. These are $$\begin{equation} \lim_{N \to \infty} \sum_{n = 1}^N \mathbb{P}(|X_n| > N) = 0 \end{equation}$$ and $$\begin{equation} \lim_{N \to \infty} \frac{1}{N^2} \sum_{n = 1}^N \mathbb{E}(|X_n|^2 I_{\{ |X_n| \leq N \}}) = \lim_{N \to \infty} \frac{1}{N} \mathbb{E}(|X_1|^2 I_{\{ |X_1| \leq N \}}) = 0 \end{equation}$$ Since $X_1, X_2, X_3, \ldots$ are identically distributed, hypothesis (ii) implies $$\begin{equation} \sum_{n = 1}^N \mathbb{P}(|X_n| > N) = \sum_{n = 1}^N \mathbb{P}(|X_1| > N) = N \mathbb{P}(|X_1| > N) \to 0 \end{equation}$$ as $N \to \infty$. This takes care of the first assumption. Denote from now on $Z_n : = X_n I_{\{ |X_n| \leq N \}}$. Result R4742: Probability calculus expression for tail probability of truncated random unsigned basic real number shows that we have $$\begin{equation} \mathbb{P}(|Z_N| > t) = \mathbb{P}(|Z_1| > t) = \left( \mathbb{P}(|X_1| > t) - \mathbb{P}(|X_1| > N) \right) I_{[0, N]} (t) \end{equation}$$ Thus, applying result R4689: Probabilistic Cavalieri principle, we have $$\begin{equation} \begin{split} \mathbb{E} |Z_N|^2 = \mathbb{E} |Z_1|^2 = \int^{\infty}_0 \mathbb{P}(|Z_1| > t) 2 t \, d t & = \int^{\infty}_0 \left( \mathbb{P}(|X_1| > t) - \mathbb{P}(|X_1| > N) \right) I_{[0, N]} (t) 2 t \, d t \\ & \leq \int^N_0 2 t \mathbb{P}(|X_1| > t) \, d t \\ \end{split} \end{equation}$$ Consider next the auxiliary function $\varphi(t) : = 2 t \mathbb{P}(|X_1| > t)$. Since $\varphi(t) \to 0$ as $t \to \infty$ and since $\varphi(t) \in [0, 2 t]$, then $\sup_{t \in \mathbb{R}} \varphi(t) < \infty$. Using result R4743: Riemann integral average of vanishing unsigned basic real function vanishes, we thus have $$\begin{equation} \begin{split} \frac{1}{N} \mathbb{E} |Z_N|^2 \leq \frac{1}{N} \int^N_0 2 t \mathbb{P}(|X_1| > t) \, d t = \frac{1}{N} \int^N_0 \varphi(t) \, d t \longrightarrow 0 \end{split} \end{equation}$$ as $N \to \infty$. This completes the proof. $\square$