Prove that $L^1\cap L^{\infty }\subseteq L^p$ for all $p\in [1,\infty]$

Question

My work:

Let $f\in L^1\cap L^{\infty }$ then $\left \| f \right \|_1,\left \| f \right \|_{\infty}$$<\infty$

$\left ( \int _X\left | f \right |^pd\mu \right )^{1/p}=\left ( \int _X\left | f \right |^{p-1}\left | f \right |d\mu \right )^{1/p}\leq \left ( \left \| f \right \|_{\infty}^{p-1}\left \| f \right \|_1 \right )^{1/p}< \infty$

Then $f\in L^p$

Correct ?

($(X,A,\mu)$ is the given measure space)

Answer 1

Yes this is correct. Maybe a step could be added after the first equality, saying that $\lvert f(x)\rvert \leqslant \lVert f\rVert_\infty$ for almost every $x$ hence $\lvert f(x)\rvert^{p-1} \leqslant \lVert f\rVert_\infty^{p-1}$ almost everywhere.