Let $(f_n)_n$ be a bounded sequence in $L^3(\mathbb R)$, such that $f_n\rightarrow f$ in $L^{3/2}(\mathbb R)$. Prove that $f_n\rightarrow f$ converges in $L^2\mathbb (R)$.
I have and idea to first use the H$\ddot {o}$lder inequality:
$\int_{\Omega} |f_n-f|^2 dx \leq ||f_n-f||_3||f_n-f||_{3/2}$
Since $3$ and ${3/2}$ are conjugates. The latter term is convergent, but $||f_n-f||_3$ had to be worked out:
$||f_n-f||_3 \leq ||f_n||_3+ ||f||_3 \ $
How do I obtain the bound for $f$ in $L^3 (\mathbb R)$ ?
I have the following idea:
$\int_\Omega |f|^3dx=\int_\Omega \lim_{n \to \infty} |f_n|^3 dx$
But how do I get the limit outside the integral or how can I use Fatou lemma here?
EDIT:I see now you had already pointed out the correct way to end. Sorry I had not read well your post(feel free to mark down my answer so to get a real one).
You can say this to get a bound also for $f$:there is a subsequence $f_{n_k}$ which converges pointwise almost everywhere to $f$. Now, using Fatou lemma you could conclude.