If the n(th) moment exists does it mean all smaller moments exist too?

Question

I would like to prove the following statement:

If the $r$th moment of a random variable $X$ exists and is finite, then all moments $1$ to $r-1$ exist and are finite.

Edit: I mean the raw moments $\mathbb E X^r$. By exists I mean exists and is finite.

Answer 1

I believe it follows from Hölder's inequality:

Answer 2

My solution lacks rigor, but here's a rough sketch:
Instead of using the concept of a moment centered around zero, such as $E(X)$ and $E(X^2)$, use the notion of a central moment, defined as $\mu_k=E\left[(X-\mu)^k\right]$.
From your statement, we know the $r$'th moment exists, so we know $\mu_r=E(X-\mu)^r$ exists. Use the binomial formula to expand that binomial, recalling that $(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k$. Then $\mu_r=E(X-\mu)^r = E\left[\sum_{k=0}^r {r \choose k}X^{k}\mu^{r-k}\right]$. Note now that this is a sum of all lower moments from $0, \dots r$, times some coefficient. Thus, in order for the r'th moment to exist, all lower moments must also exist.

Edit: There is a flaw, however, in using the central moments instead: We cannot assume that $\mu$ exists. Is there a way around that?