There exist a sequence of continuous functions $\{f_n\}$ on $\mathbb{R}$ such that $\{f_n\}$ converges to $f$ uniformly on $\mathbb{R}$, but
$$\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}f_n(x)dx \neq \int_{-\infty}^{\infty}f(x)dx$$
solution I tried-
as we know that if sequence of continuous function $f_n$ converges to a fucntion $f$ over a given interval $[a,\infty)$ then $$\lim_{n\rightarrow\infty}\int_{a}^{\infty}f_n(x)dx = \int_{a }^{\infty}f(x)dx$$ so in the above question they replace the $a$ with $-\infty$ does it effects the result?
Your method is not correct. Actually, for this one, you find a counterexample for the existence.
Consider $$f_n(x)=\frac{1}{n(1+\vert x \vert)}$$ Now try to complete the details!