The problem says :

Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable with probability density function $f$ . Suppose, for all $x \in \mathbb{R}$ , we have $f_n(x) \to f(x)$ pointwise, as $n \to \infty$ . Then, prove that : $$\lim_{n\to\infty} \int_{-\infty}^{+\infty} \Big\vert ~f_n(x) - f(x)~\Big\vert~dx ~=~ 0$$

I have already figured out a proof which does not involve anything more than the expectation version of Dominated Convergence Theorem (DCT) which can be found here in this paper. But my proof seems to be quite tedious. It'll be very good if someone can show a short and nice proof (Without using Measure Theory or the Integral version of DCT). In fact, if we go through Wikipedia, it is evident that probably the Integral version of DCT solves this problem instantly, though we haven't been taught that. However, I was thinking if there's any short and nice elementary proof.



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