Let $\{f_n\}_{n \in \mathbb{N}} \subset L^1$ be such that $f_n \rightarrow f$ in $L^1$.
And suppose that $\forall n \, \, \Vert f_n \Vert_p \le 1$ (with the usual $L^p$-norm) for some $p>1$.
Prove that $\Vert f \Vert_p \le 1$.
The hint given by the exercise is about using Fatou's Lemma, but I can't see useful ways to apply it.
This is my attempt:
$$
\int \underline{\lim} \vert f_n \vert^p \le \underline{\lim} \int \vert f_n \vert^p
= \underline{\lim} \Vert f_n \Vert^p\le 1
$$
But I'm stuck here and cannot reconduct to $f$ (I cannot deduce any pointwise convergence).
Best Answer
$\|f_n-f\|_1 \to 0$ implies there is a subsequnce $f_{n_k}$ converging a.e. to $f$. So $\int |f|^{p} \leq \lim \inf \int |f_{n_k}|^{p} \leq 1$.