I'm having trouble showing that the 2nd central moment is finite. I have $X_1,\ldots,X_n \overset{iid}{\sim} f(x)$ with $E[X_1]=\mu$ and $E[X_1^k]$ exists and is finite for any integer $k \geq 1$.
I would like to use Law of Large Numbers, so I need to show that either $E[|X_1|]$ is finite or that $E[(X_1-\mu)^2]$ is finite. I tried proving the first one with Jensen's but got stuck since absolute value is convex, not concave.
So now I'm stuck trying to show second central moment is finite.