Boundedness in $L^p$ and convergence in $L^1$ implies weak convergence in $L^p$

Let $p\in(1,\infty)$ and $f_n$ be a sequence in $\mathcal{L}^p(E)\cap\mathcal{L}^1(E)$ such that $||f_n||_p\leq1$ for every $n$ and there exist $f\in\mathcal{L}^1(E)$ such that $f_n\to f$ in $\mathcal{L}^1(E)$ norm.Prove that $f\in\mathcal{L}^p(E)$ and that $f_n \rightharpoonup f$ weakly in $\mathcal{L}^p(E)$.I don't know how to prove either, assuming $f\in\mathcal{L}^p(E)$ i have proved that $f_n\to f$ in $\mathcal{L}^r(E)$ norm for $r\in(1,p)$ using Holder inequality, maybe this is useful. I also know that strong convergence in $\mathcal{L}^1(E)$ norm, implies the existence of a subsequence converging pointwise a.e., but can't think how to use this fact.(I'm working with the lebesgue measure, $E\subseteq\mathbb{R}^n$ measurable.)