I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$.
To show: If $f \in L_1([0,1])$ is such that $||f_n - f||_1 \rightarrow 0$, then $||f||_p < \infty$.
I haven't managed to start this off. Note that $f_n$ might not converge to $f$ in $L_p$, e.g. take $\sqrt{n} \cdot \chi_{[0, \frac{1}{n}]}$. Can somebody give me the first step or two? I'd really appreciate it.
Thanks!
It is possible to choose a subsequence $\left(f_{n_j}\right)_{j\geqslant 1}$ which converges to $0$ almost everywhere. Applying Fatou's lemma to the non-negative sequence $\left(\left\lvert f_{n_j}\right\rvert^p \right)_{j\geqslant 1}$, we get $$\int \left\lvert f\right\rvert^p=\int\liminf_{j\to +\infty} \left\lvert f_{n_j}\right\rvert^p \leqslant \liminf_{j\to +\infty}\int \left\lvert f_{n_j}\right\rvert^p \leqslant\sup_{i}\left\lVert f_i\right\rVert^p_p < +\infty.$$