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

(i)	\begin{equation} \forall \, n \in 1, 2, 3, \ldots : \mathbb{E} |X_n|^2 < \infty \end{equation}
(ii)	$\lambda_1, \lambda_2, \lambda_3, \ldots \in \mathbb{R}$ are each a D993: Real number
(iii)	\begin{equation} \lim_{n \to \infty} \frac{\mathsf{Var} X_n}{\lambda^2_n} = 0 \end{equation}

By hypothesis (iii), we have \begin{equation} \mathbb{E} \left| \frac{X_n - \mathbb{E} X_n}{\lambda_n} \right| = \frac{1}{\lambda^2_n} \mathbb{E} |X_n - \mathbb{E} X_n|^2 = \frac{1}{\lambda^2_n} \mathsf{Var} X_n \longrightarrow 0 \end{equation} as $n \to \infty$. This shows that $(X_n - \mathbb{E} X_n) / \lambda_n \overset{L^2}{\longrightarrow} 0$ and the result is now a consequence of R3250: P-convergence implies convergence in probability. $\square$