Let be $f:[0,\infty[ \rightarrow \mathbb{R}$ continuous. If $\displaystyle \int_0^{\infty} f(x) dx $ converges so $\displaystyle \lim_{r\rightarrow \infty} \int_r^{\infty} f(x) dx=0$
My Attempt:
Call $G(r) = \displaystyle \int_0^{r} f(x)dx, \hspace{0.2cm}$ then $ \hspace{0.2cm}\displaystyle \lim_{r\rightarrow \infty} G(r) = L$
and we have $L = \displaystyle \lim_{r\rightarrow \infty} G(r) = \displaystyle \lim_{r\rightarrow \infty} \displaystyle \int_0^{\infty} f(x) dx = \lim_{r\rightarrow \infty} \displaystyle \int_0^{r} f(x) dx$ + $\displaystyle \int_r^{\infty} f(x) dx = \lim_{r\rightarrow \infty} \displaystyle \int_0^{r} f(x) dx + \lim_{r\rightarrow \infty} \displaystyle \int_r^{\infty} f(x) dx = L + \lim_{r\rightarrow \infty} \displaystyle \int_r^{\infty} f(x) dx $
Then $\displaystyle \lim_{r\rightarrow \infty} \displaystyle \int_r^{\infty} f(x) dx= L-L = 0$
But I think that this isn't correct because this expression $ \displaystyle\lim_{r\rightarrow \infty} \displaystyle \int_0^{r} f(x) dx + \lim_{r\rightarrow \infty} \displaystyle \int_r^{\infty} f(x) dx $,
Is there a way to change this expression or another way to solve this exercise?