Ok, I think you mean $\displaystyle f_ng_n\rightarrow fg$ in the second one.
For the first one, write
$\displaystyle f_n^2=(f_n-f+f)^2=(f_n-f)^2+2(f_n-f)f+f^2$
and note that $\displaystyle f_n-f\rightarrow 0$ in measure.
For the second one, note
$\displaystyle (f_n+g_n)^2\rightarrow (f+g)^2$
by part 1, and expand the squares.