$f_n$ converges to $f$ in $L^3(X,\mathcal{M},\mu)$ , prove that $f_n^3$ converges to $f^3$ in $L^1(X,\mathcal{M},\mu) $

Question

Suppose $(X,\mathcal{M},\mu)$ is a complete measure space. If $f_n$ converges to $f$ in $L^3(X,\mathcal{M},\mu)$ , prove that $f_n^3$ converges to $f^3$ in $ L^1 (X,\mathcal{M},\mu) $.

Actually my issue is the following :

$$\int_X |f^3-f_n^3|d\mu=\int_X |f-f_n||f^2+ff_n+f_n^2|d\mu$$ If I know that $f^2+ff_n+f_n^2$ belongs to $L^{3/2}(X,\mathcal{M},\mu)$ I can use the Holders inequality and I am done (in fact I need to show that $ff_n \in L^{3/2}(X,\mathcal{M},\mu)$) I'd appreciate your hints/answers

Answer 1

in fact I need to show that $ff_n \in L^{3/2}(X,\mathcal{M},\mu)$

This is Cauchy-Schwarz: $$ \left(\int\left|ff_n\right|^{3/2}\mathrm d\mu\right)^2\leqslant\int\left|f\right|^{3}\mathrm d\mu\cdot\int\left|f_n\right|^{3}\mathrm d\mu. $$