I tried to prove the following
Theorem: Given $(X_n)_{n\in\mathbb{N}}$ iid. random variables with $\mathbb{E}[X_i^2]<\infty$. If the rv's have respective densities $(f_n)_{n\in\mathbb{N}}$ and $f_n\rightarrow f$ pointwise, it follows that $X_n\stackrel{d}{\rightarrow}X$, meaning convergence in distribution.
Proof: $$|F_n(x)-F(x)|\le \left | \int_{(-\infty, x]}f_n(t)\,\mathrm{d}t - \int_{(-\infty, x]}f(t)\,\mathrm{d}t\right |\\ \le \int_{(-\infty, x]}|f_n(t)-f(t)|\,\mathrm{d}t\le \int_\mathbb{R}|f_n(t)-f(t)|\,\mathrm{d}t$$
Now because of Scheffés-Lemma I know that the rhs. converges to zero.
Is my proof correct? I did not use the finite variance which gives me doubt.
To apply Scheffé's lemma, you must prove that $$\int_{\mathbb{R}} f_n \to \int_{\mathbb{R}} f$$
However, your proof doesn't show why this should be the case.