Assume $P_n, n\in\mathbb{N}$ and $P$ are absolutely continuous probability measures with respect to a sigma finite measure $\mu$ on $(\mathbb{R},\mathcal{B})$. Let $f_n, n\in \mathbb{N}$ and $f$ be the densities of above measures respectively.
I want to prove that if $f_n$ converges pointwise to $f$, then $P_n$ converges weakly to $P$. Here is what I have tried so far. I've read that even strong (pointwise) convergence follows, so I tried to prove that and the weak convergence follows:
Let $A \in \mathcal{B}$ and since $f_n$ are the the densities of $P_n$the following equalities hold $$P_n(A) = \int\limits_{A} f_n d\mu = \int 1_{A} f_n d\mu.$$ Now since $f_n$ converges pointwise to $f$, also $1_{A} f_n$ converges pointwise to $1_{A} f$. Here is where I'm stuck because I want to use the theorem about dominated convergence. Hence, I have to find an integrable bound for $1_{A} f_n$. If the theorem about dominated convergence applies I get
$$\lim\limits_{n\rightarrow\infty}P_n(A)=\int \lim\limits_{n\rightarrow\infty} 1_{A} f_n d\mu = \int 1_{A} f d\mu=\int\limits_{A} f d\mu = P(A)$$ which completes the proof.
Can someone help me to find a bound for $f_n$? Is this approach correct or am I missing something?