Let $f\in L^{p_{1}}$. We need to show that $f \in L^{p_{2}}$ in a finite measure space. Where $p_{1} \geq p_{2}$.
My attempt:
$\int |f|^{p_{2}}=\int_{|f|<1} |f|^{p_{2}} + \int_{|f|\geq 1} |f|^{p_{2}}$
$ \leq \mu(X) + \int_{|f|\geq 1} |f|^{p_{1}} < \infty$.
Thus $f \in L^{p_{2}}$.
Is this correct?