If for some $A$, $\int\_A^\infty f(x)dx$ is convergent then $\int\_a^\infty f(x)dx$ ($a<A$) is also convergent and $\int\_a^\infty f(x)dx=\int\_a^A f(x)dx + \int\_A^\infty f(x)dx$

We know that $\int\_a^B f(x)dx=\int\_a ^Af(x) + \int\_A^Bf(x)dx$

Because $f(x)$ is integrable on $[c,d]$ $\int\_a ^Af(x)dx$ is finite.We have $\int\_A^\infty f(x)dx$ is convergent so if we take limit as $B\to +\infty$ we get

$\int\_a^\infty f(x)dx=\int\_a^Af(x)+\int\_A^\infty f(x)dx$.What I did is correct?Sorry if this seems silly question improper integrals are new chapter for me.

EDIT.

In order $\int\_a^\infty f(x)dx$ to be convergent $\iff$ for some $A$ $\int\_A ^\infty f(x)dx$ to be convergent.