During my studies for intro. to probability (undergrad) we learned about Jensen's inequality. The lecturer showed as the following theorem during one of our lectures:
Suppose $Y$ is a random variable with $E(|Y|^\alpha)<\infty$ for some $\alpha > 0$, Then $E(|Y|^\beta)<\infty$ for $0\leq \beta \leq \alpha$
I'm attempting to prove this theorem. I've tried using the definition of expectation and Jensen's inequality, by showing that the series converges (I've also tried absolute convergence). But I'm not sure which convex function I should choose, and all of my attempts were dead ends.
Any suggestions will be very helpful.