How can I prove that $f\in L^p\cap L^q\implies f\in L^r$ for all $r\in [p,q]$?

Question

I have a result that says that since $f\in L^p(\mathbb R)\cap L^q(\mathbb R),$ we have that $f\in L^r(\mathbb R)$ for all $r\in [p,q]$. I don't really know how to prove this. I know that if $r\in [p,q]$, then $$\frac{1}{r}=t\frac{1}{p}+(1-t)\frac{1}{q},$$ for a certain $t\in [0,1]$. Then, I tried to apply Holder at $\int|f|^r$ but I don't really see how.

Answer 1

$\int |f|^{r} =\int_{\{|f| \leq 1\}} |f|^{r} +\int_{\{|f| > 1\}} |f|^{r}$. The first term does not exceed $\int_{\{|f| \leq 1\}} |f|^{p}$ and the first term does not exceed $\int_{\{|f| \leq 1\}} |f|^{q}$.

Answer 2

Hint

Using your notations, apply Holder to $$\int|f|=\int|f|^{1-t}|f|^t.$$