Let $1\leq p < r < q \leq \infty$ and $E\in \mathbb{R}$. Define $$A = L^p(E) + L^q(E) = \{f=g+h:g\in L^p(E), h\in L^q(E) \}$$
and $$\|f\|_A = \inf_{f=g+h} \|g\|_p+\|h\|_q$$ where the infimum is taken over all decompositions of $f$.
I already showed that $L^r(E) \subset A$ and that $\|f\|_A \leq \|f\|_r^{r/p} + \|f\|_r^{r/q}$, and now i must use this to prove that the injection of $L^r$ into $A$ is continuous to conclude that there exist $C>0$ such that $$\|f\|_A \leq C\|f\|_r.$$ Any hint?