Here is the problem:

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> Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$
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I'm almost sure that this is a simple exercise on Hölder's inequality yet I can't find the right $a,b>1$ with $$\frac{1}{a}+\frac{1}{b}=1$$

to apply the inequality.

Whether you'll give a hint or a full solution (which I expect to be very short), please explain your thought process. I guess it's like with the $\varepsilon/\delta$-proofs where things happen to appear out of the blue but we know that it's because of some scratch-work before.