How to use an interpolation argument to prove $A\hookrightarrow L^p(\mathbb R)$ for $p\in [2, +\infty]?$

Question

Assume you have that $$A\hookrightarrow L^2(\mathbb R) \quad\text{ and }\quad B\hookrightarrow L^\infty(\mathbb R).$$ Suppose also that $A\hookrightarrow B$. By using the above information, is it possible to prove that $$A\hookrightarrow L^p(\mathbb R)\quad\mbox{ for all } p\in [2, +\infty]?$$

I asked the lecturer, and he said that it can be proved by using an interpolation argument. What does it mean? Anyone could please help me with the proof?