Proving Fatou type lemma Let $f_1, f_2, \cdots$ and $f$ be nonnegative lebesgue integrable functions on $\mathbb{R}$ such that $$\lim_{n \to \infty}\int_{-\infty}^y f_n(x)dx = \int_{-\infty}^y f(x)dx   \; \; \text{ for each $y \in \mathbb{R}$}$$ and $$\lim_{n \to \infty}\int_{-\infty}^{\infty} f_n(x)dx = \int_{-\infty}^{\infty} f(x)dx $$ 
Then I want to prove that $ \liminf_{n \to \infty} \int_{U}f_n(x)dx  \geq \int_{U}f(x)dx\:$ for any open set $U$ of $\mathbb{R}.$
I think this can done by proving $\lim_{n \to \infty}f_n(x) = f(x) $ and then using Fatou's lemma. But I am not able to prove $\lim_{n \to \infty}f_n(x) = f(x) $ .
Thank you in advance.

[But I am not able to prove $\lim\limits_{n\to \infty} f_n(x)=f(x)$], as said by Ramiro and Sir Gerard, it may be false. For example 
Suppose that $g(x)$ is a integrable function which is defined on $[1,2]$. We can denote that
 $f(x) = g(x)$ if $x \in [1,2]$, else $f(x) = 0$. Then $\int\limits_U g(x) dx = \int\limits_U f(x)$ for any $U \subset \mathbb{R}$. Now, for any $k \in \mathbb{N}_+$, take
$f_k(x) = 1 $ if $x \in [0, 1/k]$, else, $f_k(x) = f(x)$. Then $\lim\limits_{k \to + \infty} \int\limits_{-\infty}^{y} f_k(x)dx = \int\limits_{-\infty}^y f(x) dx$ for any $y \in \mathbb{R} \cup \{+ \infty\}$.
But $\lim\limits_{k \to + \infty} f_k(x) \neq f(x)$ because $\lim\limits_{k \to +\infty} f_k(x)  =1$ at $x = 0$.  
