If $f$ is Lebesgue integrable on a measurable set $E$, then is $f$ Lebesgue integrable on any measurable subset $A$ of $E$?

Question

I am not sure but I think the statement is true because
$f=f^+-f^-$, where $f^+$ and $f^-$ are non negative functions. We know that $f$ is Lebesgue integrable iff both $f^+$ and $f^-$ are Lebesgue integrable and so, $\int_Ef^+$ and $\int_Ef^-$ are finite. Now since $f^+$ and $f^-$ are non negative so for any measurable subset $A$ of E, $\int_Af^+\le \int_Ef^+$ and $\int_Af^-\le \int_Ef^-$ and so $\int_Af^+$ and $\int_Af^-$ are finite which gives $\int_Af$ is finite i.e. $f$ is Lebesgue integrable over A.

Is my answer correct?