The convergence of a r.v. moment of order r, about a and its central moments

Question

$r>0$, $E[|X|^r] < \infty$ $\ $ iff $\ $ $E[|X-a|^r] < \infty$ $\ $ for every $a$.
It is trivial when $r=2$, but how could I proof the statement with the other values of $r$?

Answer 1

$|X-a|^{r} \leq (|X|+|a|)^{r}\leq (2 \max \{|X|,|a|\})^{r}= 2^{r} \max \{|X|^{r},|a|^{r}\}\leq 2^{r} (|X|^{r}+|a|^{r})$ so $E|X-a|^{r} \leq 2^{r} (E|X|^{r}+|a|^{r})$; similar argument in the other direction.