[proofplan] We prove contiguity directly from the definition. Given events $A_n$ with $P_n(A_n)\to 0$, we rewrite $Q_n(A_n)$ as a $P_n$-expectation using the Radon--Nikodym derivative $L_n$. An $L^2(P_n)$ estimate bounds this expectation by the square root of $P_n(A_n)$ times the uniformly bounded second moment of $L_n$, forcing $Q_n(A_n)\to 0$. [/proofplan]

[step:Rewrite each $Q_n(A_n)$ as a $P_n$-expectation] Let $(A_n)_{n\in\mathbb{N}}$ be a sequence with $A_n \in \mathcal{F}_n$ for each $n\in\mathbb{N}$ and $P_n(A_n)\to 0$. For each $n$, define the indicator function $\mathbb{1}_{A_n}: \Omega_n \to \{0,1\}$ by setting $\mathbb{1}_{A_n}(\omega)=1$ when $\omega \in A_n$ and $\mathbb{1}_{A_n}(\omega)=0$ when $\omega \notin A_n$. Since $L_n=dQ_n/dP_n$, the defining property of the [Radon--Nikodym derivative](/page/Radon-Nikodym%20Derivative) gives \begin{align*} Q_n(A_n) = \int_{\Omega_n} \mathbb{1}_{A_n}(\omega)\,dQ_n(\omega) = \int_{\Omega_n} L_n(\omega)\mathbb{1}_{A_n}(\omega)\,dP_n(\omega) = \mathbb{E}_{P_n}[L_n\mathbb{1}_{A_n}]. \end{align*} [/step]

[step:Control the likelihood mass of $A_n$ by the second moment]For each $n$, set \begin{align*} a_n := \mathbb{E}_{P_n}[L_n^2], \qquad b_n := P_n(A_n), \qquad c_n := \mathbb{E}_{P_n}[L_n\mathbb{1}_{A_n}]. \end{align*} The [Cauchy--Schwarz inequality](/page/Cauchy--Schwarz%20Inequality) in $L^2(\Omega_n,\mathcal{F}_n,P_n)$ applied to the functions $L_n$ and $\mathbb{1}_{A_n}$ gives \begin{align*} c_n^2 \leq a_n\,\mathbb{E}_{P_n}[\mathbb{1}_{A_n}^2] = a_n\,P_n(A_n) = a_n b_n. \end{align*} The equality $\mathbb{E}_{P_n}[\mathbb{1}_{A_n}^2]=P_n(A_n)$ follows from $\mathbb{1}_{A_n}^2=\mathbb{1}_{A_n}$.[/step]

[guided]The point of the second-moment hypothesis is that it turns small $P_n$-probability into small $Q_n$-probability. After the Radon--Nikodym rewrite, the relevant quantity is \begin{align*} Q_n(A_n)=\mathbb{E}_{P_n}[L_n\mathbb{1}_{A_n}]. \end{align*} This expectation is the $P_n$-average of the likelihood ratio $L_n$ over the event $A_n$. We now estimate it in $L^2(P_n)$. For each $n$, define \begin{align*} a_n := \mathbb{E}_{P_n}[L_n^2], \qquad b_n := P_n(A_n), \qquad c_n := \mathbb{E}_{P_n}[L_n\mathbb{1}_{A_n}]. \end{align*} The [Cauchy--Schwarz inequality](/page/Cauchy--Schwarz%20Inequality) for the product of two square-integrable functions in $L^2(\Omega_n,\mathcal{F}_n,P_n)$ gives \begin{align*} \left(\mathbb{E}_{P_n}[L_n\mathbb{1}_{A_n}]\right)^2 \leq \mathbb{E}_{P_n}[L_n^2]\, \mathbb{E}_{P_n}[\mathbb{1}_{A_n}^2]. \end{align*} Here $L_n\in L^2(P_n)$ by the finite second-moment assumption, and $\mathbb{1}_{A_n}\in L^2(P_n)$ because $P_n$ is a probability measure. Since $\mathbb{1}_{A_n}^2=\mathbb{1}_{A_n}$ pointwise, \begin{align*} \mathbb{E}_{P_n}[\mathbb{1}_{A_n}^2] = \mathbb{E}_{P_n}[\mathbb{1}_{A_n}] = P_n(A_n). \end{align*} Therefore \begin{align*} Q_n(A_n)^2 = \left(\mathbb{E}_{P_n}[L_n\mathbb{1}_{A_n}]\right)^2 \leq \mathbb{E}_{P_n}[L_n^2]\,P_n(A_n). \end{align*} This is the whole mechanism: a uniform bound on $\mathbb{E}_{P_n}[L_n^2]$ prevents $Q_n$ from placing asymptotically non-negligible mass on sets whose $P_n$-mass vanishes.[/guided]

[step:Use the uniform second-moment bound to conclude contiguity] Define \begin{align*} C := \sup_{n\in\mathbb{N}} \mathbb{E}_{P_n}[L_n^2]. \end{align*} By hypothesis, $C<\infty$. Combining the previous steps, for every $n\in\mathbb{N}$, \begin{align*} 0 \leq Q_n(A_n)^2 \leq C\,P_n(A_n). \end{align*} Since $P_n(A_n)\to 0$, the [squeeze theorem](/theorems/627) gives $Q_n(A_n)^2\to 0$, and because $Q_n(A_n)\geq 0$, it follows that $Q_n(A_n)\to 0$. Thus every sequence of events whose $P_n$-probabilities vanish also has vanishing $Q_n$-probabilities. This is precisely [contiguity](/page/Contiguity) of $(Q_n)_{n\in\mathbb{N}}$ with respect to $(P_n)_{n\in\mathbb{N}}$. [/step]