I have a problem with following statement: We have $f \in W^{1,1}(B) (B-ball\ in\ \mathbb{R^n}), \ \nabla f \ e^{f} - $ integrable $\Rightarrow \ e^f \in W^{1,1}$
I've started with a sequence of functions $f_n \in C^1(B) \bigcap W^{1,1}(B)$ and $f_n \rightarrow f$ in $W^{1,1}(B)$. Next by Fatou's lemma we have $$\int\limits_B e^f dx \le \lim\inf\limits_{n \rightarrow \infty} \int\limits_{B} e^{f_n} dx$$ But why right side of inequality have a limit?
I think that I should use $f_n$ such that $$f_n (x)= \left\{\begin{array}\ n, \ f(x)>n \\ f, \ |f(x)|\le n \\ -n, \ f(x)<-n \end{array}\right.$$ So we have that \begin{equation} \nabla f_n (x)= \left\{\begin{array}\ 0, \ f(x)>n \\ \nabla f, \ |f(x)|\le n \\ 0, \ f(x)<-n \end{array}\right. \end{equation} And $f_n\in W^{1,1}(B), \ f_n \rightarrow f \ \in W^{1,1}(B)$
I have to prove that $e^{f_n} \in W^{1,1}(B)$. But this follows from $e^{f_n}\le e^n$ and $|\nabla f_n| e^{f_n}\le |\nabla f|e^f$ .
Then I should use Poincare inequality for proving that $\lim\int\limits_{B}e^{f_n}dx<+\infty$
Am I right?