Let $\Omega=(0,1)$ and $f_n(x)=ne^{-nx}$ prove that
(i) $f_n\rightarrow0$ a.e.
(ii) $f_n$ is bounded in $L_1(\Omega)$
I know that for $f_n$ to converge a.e. to zero, the set of points in which this does not happen must have zero measure i think it has something to do with the interval $\left[0,\dfrac{1}{2n}\right]$ but I have no idea how to proceed. I don't know what to do with (ii)
Thanks in advance for any help.