Prove that $\forall a \in \mathbb{R} \ , \mathbb{E}\left\{|\boldsymbol{X}-\boldsymbol{a}|^{p}\right\}<\infty$ with conditions.

Question

Let $ p\geqslant1 $ and b be a real number. Having $\mathbb{E}\left\{|\boldsymbol{X}-\boldsymbol{b}|^{p}\right\}<\infty$, I would like to prove that $$\forall a \in \mathbb{R} \ \ \ , \mathbb{E}\left\{|\boldsymbol{X}-\boldsymbol{a}|^{p}\right\}<\infty$$

I thought of using the moment generating function. But I can't seem to see a path to the solution with that idea.

Any suggestions??

Answer 1

$HINT$

$$|X-a|^p\leq (|X-b|+|b-a|)^p$$ $$ \leq2^p\max\{|X-b|^p,|b-a|^p\} \leq 2^p(|X-b|^p+|b-a|^p)$$