Weak convergence and weak star convergence.

If region $\Omega$ is bounded and $u_n$ has weak star convergence in $L^\infty ( \Omega)$ to some $u\in L^\infty(\Omega)$ , does it imply that $u_n$ converges weakly in any $L^p(\Omega) $ ?

I think i got it : If $sup$ of a function is finite then integral over a bounded region is finite with any $p$ norm . is it right ?

$\{u_n\}\subset L^\infty(\Omega)$ converges in the weak star topology to $u\in L^\infty(\Omega)$ if $$ \lim_{n\to\infty}\int_\Omega u_n\phi\,dx=\int_\Omega u\,\phi\,dx\quad\forall\phi\in L^1(\Omega). $$ Since $\Omega$ is bounded, $L^\infty(\Omega)\subset L^p(\Omega)\subset L^1(\Omega)$ for all $p\ge1$. It follows that $u_n$ converges weakly to $u$ in $L^p(\Omega)$ for all $p\in[1,\infty)$.