I would like to prove that If $E[|X|^n] < \infty$ for some positive integer $n$, then $E[|X|^m] < \infty$ for all positive integers $m ≤ n$. So far, using Jensen's Inequality, I have that $$ \infty > E[|X|^n] \geq |E[X]|^n \geq |E[X]|^m. $$
From here, I'm stuck... Any help is greatly appreciated.
Suppose that $n > m$. Use Holder's inequality with $p = n/m$ and $q = (1-p^{-1})^{-1}$, so that $p^{-1} +q^{-1} = 1$, instead. $$E[|X|^m] = E[|X|^m \cdot 1] \le E[|X^m|^p]^{1/p} \, E[1^q]^{1/q} = E[|X|^n]^{m/n} < \infty$$ since $E[|X|^n] < \infty$.