a problem about convergence in $L^{P}$ space

Question

I need a small certain hint for solving this problem please.

Problem: Let $(f_{n})_{n} $ be a bounded sequence in $ L^{3}(\mathbb{R}), $ such that $ f_{n}\to f $ in $ L^{\frac{3}{2}}(\mathbb{R}). $ Prove that $ f_{n}\to f $ in $ L^{2}(\mathbb{R}). $

Answer 1

Due to holder inequality:$$\int|f||g|dx\leq\left(\int|f|^pdx\right)^{1/p}\left(\int|g|^qdx\right)^{1/q}$$ So if $p= 3$ and $q= \frac{3}{2}$, we get $$\int|f_n-f||f_n-f|dx\leq\left(\int|f_n-f|^{3}dx\right)^{1/3}\left(\int|f_n-f|^{3/2}dx\right)^{2/3}.$$ Since $f_n$ is bounded in $L^3(\mathbb{R})$, $\Vert f_n-f\Vert_{L^3(\mathbb{R})}(\leq\Vert f_n\Vert+\Vert f\Vert)$ is bounded too. In the meantime, $f_n\to f$ in $L^{3/2}(\mathbb{R})$, so $$\left(\int|f_n-f|^{3/2}\right)^{2/3}dx\to0 \quad as \quad n\to\infty$$ All in all, we'll see$$\int|f_n-f||f_n-f|dx\to0 \quad as \quad n\to\infty$$

P.S. Thanks to @zhw.

since $f_n\to f$ in $L^{3/2}(\mathbb{R})$ , then we have $f_n\xrightarrow{m} f$ in $\mathbb{R}$. According to Riesz theorem, there is a subsequence of $\{f_n\}_n$, named as $\{f_{n_k}\}_k$, that converges to $f$ almost everywhere in $\mathbb{R}$.

Due to Fatou's lemma $$\int|f|^3dx=\int \varliminf|f_{n_k}|^3dx\leq \varliminf\int|f_{n_k}|^3dx<\infty$$