Prove that $\cup_{1<p\leq\infty} L^p([0,1])\subset L^1([0,1]) $.
Let $f\in\cup_{1<p\leq\infty}L^p([0,1])$.
Then $f\in L^p([0,1])$ for some $p$. We write $\int|f|d\mu < (\int|f|^p\ )^\frac{1}{p}(\int1^q\ )^\frac{1}{q}$ since $\int|f|^p$ is finite
then $\int|f|d\mu$ is finite. Therefore $f\in L^1([0,1] )$
I feel like this proof is too simple. there should be more cases. Is this explanation enough to prove this claim?