Weak star convergence in $L^{\infty}$ and strong convergence in $L^2$ imply strong convergence in $L^p$

Question

I can't see why the following would hold. Can someone please point out why?

If $f_n$ convergence weak-* in $L^{\infty}(\Omega)$, and converges strongly in $L^2(\Omega)$, then it must also converge strongly in $L^p(\Omega)$ for $1 \leq p < \infty$

Answer 1

Hint: A standard application of Hölder's inequality gives $$ \|g\|_{L^p} \le \|g\|_{L^2}^\theta \|g\|_{L^\infty}^{1-\theta}$$ for a proper choice of $\theta$.