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. $\square$
