Basic fact in $L^p$ space I'm studying $L^p$ space.
$1 \le p < r <q < \infty$ then $L^p \cap  L^q \subset L^r$.
More over $L^p \cap  L^\infty \subset L^r$
I'm trying to prove that fact. Which theorem is useful for proving that?
 A: (1.) For every positive real number $a$, one has $a^r\leqslant a^p+a^q$ (one can prove separately the cases $a\leqslant1$ and $a\gt1$). Thus, $|f(x)|^r\leqslant |f(x)|^p+|f(x)|^q$ for every $x$. 
Hence $\|f\|_r^r\leqslant\|f\|_p^p+\|f\|_q^q$ and $\|f\|_r$ is finite for every $f$ in $L^p\cap L^q$. Thus, $L^p\cap L^q\subset L^r$.
(2.) For every positive real numbers $a$ and $b$ such that $a\leqslant b$, one has $a^r\leqslant a^pb^{r-p}$. Thus, $|f(x)|^r\leqslant |f(x)|^p\cdot\|f\|_\infty^{r-p}$ for every $x$. 
Hence $\|f\|_r^r\leqslant\|f\|_p^p\cdot\|f\|_\infty^{r-p}$ and $\|f\|_r$ is finite for every $f$ in $L^p\cap L^\infty$. Thus, $L^p\cap L^\infty\subset L^r$.
Note: Although such uniform pointwise inequalities cannot yield optimal norm inequalities, they are (i) simple to prove, and (ii) sufficient to get the inclusions of spaces the OP is interested in.
A: For $f(x)\ge0$, Jensen's Inequality yields
$$
\left(\frac{1}{\int_X f^p(x)\,\mathrm{d}x}\int_X f^{r-p}(x)f^p(x)\,\mathrm{d}x\right)^{\Large\frac{q-p}{r-p}}\le\frac{1}{\int_X f^p(x)\,\mathrm{d}x}\int_X f^{q-p}(x)f^p(x)\,\mathrm{d}x
$$
which becomes
$$
\left(\int_Xf^r(x)\,\mathrm{d}x\right)^{\Large\frac1r}\le\left(\int_Xf^p(x)\,\mathrm{d}x\right)^{\Large\frac1p\left(\frac{p}{r}\frac{q-r}{q-p}\right)}\left(\int_X f^q(x)\,\mathrm{d}x\right)^{\Large\frac1q\left(\frac{q}{r}\frac{r-p}{q-p}\right)}
$$
Thus for $f\in L^p\cap L^q$,
$$
\|f\|_r\le\|f\|_p^{\Large\frac{p}{r}\frac{q-r}{q-p}}\;\|f\|_q^{\Large\frac{q}{r}\frac{r-p}{q-p}}\tag{1}
$$
where
$$
\frac{p}{r}\frac{q-r}{q-p}+\frac{q}{r}\frac{r-p}{q-p}=1\tag{2}
$$
Note that when $q\to\infty$, $(1)$ becomes
$$
\|f\|_r\le\|f\|_p^{\Large\frac{p}{r}}\;\|f\|_\infty^{\Large1-\frac{p}{r}}\tag{3}
$$
A: Some hints:
$(1.)$ For $L^{p}\cap L^{q}\subset L^{r}$, take and $f\in L^{p}\cap L^{q}$ and divide the integration domain $X$ to $A:=\{x\in X:|f(x)|\leq 1\}$ and $A^{c}$. What can you say about $|f(x)|^{r}$ for $x\in A$ or $x\in A^{c}$? What can you conclude for $\|f\|_{r}$?
$(2.)$ Start showing that if $f\in L^{p}$ then $\mu(A^{c})<\infty$. Then use the fact that $|f(x)|\leq \|f\|_{\infty}$ for $\mu$-a.e. $x\in X$ and use the same logic as in the previous step to conclude $\|f\|_{r}<\infty$ if $f\in L^{p}\cap L^{\infty}$. 
If you need some more hints or can't get started with these; we can discuss at the comment section below.
